Copyright by Yongcun Feng 2016
Transcript of Copyright by Yongcun Feng 2016
The Dissertation Committee for Yongcun Feng Certifies that this is the approved
version of the following dissertation:
Fracture Analysis for Lost Circulation and Wellbore Strengthening
Committee:
Kenneth E. Gray, Supervisor
Hugh C. Daigle
John T. Foster
John F. Jones
Mark W. McClure
Fracture Analysis for Lost Circulation and Wellbore Strengthening
by
Yongcun Feng, B.E.; M.E.
Dissertation
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Fulfillment
of the Requirements
for the Degree of
Doctor of Philosophy
The University of Texas at Austin
August 2016
Dedication
To my parents, Jiatang Feng and Xiuhong Zhang,
To my wife, Xiaorong Li,
for their endless love, support and encouragement.
v
Acknowledgements
It is impossible to complete this dissertation without the help from a number of
people. First of all, I would like to express my sincerest gratitude to my supervisor, Dr.
Kenneth E. Gray, for his inspiration, encouragement, and guidance throughout my study at
The University of Texas at Austin. His friendly character and professional supervision have
made my study at UT-Austin fruitful and enjoyable. I would also like to thank my
dissertation committee members, Dr. Hugh C. Daigle, Dr. John T. Foster, Mr. John F.
Jones, and Dr. Mark W. McClure, for taking time to review my dissertation and providing
me with valuable feedback and comments. Thanks are also extended to Dr. Evgeny G.
Podnos for helping me get started with Abaqus®.
I wish to thank the Wider Windows Industrial Affiliate Program, the University of
Texas at Austin, for financial support of this dissertation and my graduate program. Project
support and technical discussions with industrial colleagues from Wider Windows
sponsors BHP Billiton, British Petroleum, Chevron, ConocoPhillips, Halliburton,
Marathon, National Oilwell Varco, Occidental Oil and Gas, and Shell are gratefully
acknowledged.
I would like to thank my colleagues in the Wider Windows Industrial Affiliate
Program for their support in my study and life at UT-Austin: Yao Fu, Anthony Ho, Zhi Ye,
Cesar Soares, Arjang Gandomkar, Chiranth Hegde, Peidong Zhao, Yangyang Chen,
vi
Hamza Jaffal, Lucas Barros, Scott Wallace, Tyler Adams, Xuyue Chen, Chao Gao, and
Bishwas Ghimire.
I also would like to express my gratitude to the following friends for their friendship
and help, which gave me a wonderful life with numerous good memories during my Ph.D.
study at UT-Austin: Bo Ren, Xiaoning Tan, Guang Yang, Yun Wu, Weiwei Wu, Huizi
Han, Haotian Wang, Yu Liang, Chunbi Jiang, Tianbo Liang, Hao Pang, Kai Zhang, and
Nineng Xu.
Finally, thank you to my parents, Jiatang Feng and Xiuhong Zhang, for their endless
love, encouragement, and support. My special gratitude is to my wife Xiaorong Li for her
love, understanding and encouragement all the time in my life.
vii
Fracture Analysis for Lost Circulation and Wellbore Strengthening
Yongcun Feng, Ph.D.
The University of Texas at Austin, 2016
Supervisor: Kenneth E. Gray
Lost circulation is the partial or complete loss of drilling fluid into a formation. It
is among the major non-productive time events in drilling operations. Most of the lost
circulation events are fracture initiation and propagation problems, occurring when fluid
pressure in a wellbore is high enough to create fractures in a formation. Wellbore
strengthening is a common method to prevent or remedy lost circulation problems.
Although a number of successful field applications have been reported, the fundamental
mechanisms of wellbore strengthening are still not fully understood. There is still a lack of
functional models in the drilling industry that can sufficiently describe fracture behavior in
lost circulation events and wellbore strengthening.
A finite-element framework was first developed to simulate lost circulation while
drilling. Fluid circulation in the well and fracture propagation in the formation were
coupled to predict dynamic fluid loss and fracture geometry evolution in lost circulation
events. The model provides a novel way to simulate fluid loss during drilling when the
boundary condition at the fracture mouth is neither a constant flowrate nor a constant
pressure, but rather a dynamic wellbore pressure.
There are two common wellbore strengthening treatments, namely, preventive
treatments based on plastering wellbore wall with mudcake before fractures occur and
remedial treatments based on bridging/plugging lost circulation fractures. For preventive
viii
treatments, an analytical solution and a numerical finite-element model were developed to
investigate the role of mudcake. Transient effects of mudcake buildup and permeability
change on wellbore stress were analyzed. For remedial treatments, an analytical solution
and a finite-element model were also proposed to model fracture bridging. The analytical
solution directly predicts fracture pressure change before and after fracture bridging; while
the finite-element model provides detailed local stress and displacement information in
remedial wellbore strengthening treatments.
In this dissertation, a systematic study on lost circulation and wellbore
strengthening was performed. The models developed and analyses conducted in this
dissertation present a useful step towards understanding of the fundamentals of lost
circulation and wellbore strengthening, and provide improved guidance for lost circulation
prevention and remediation.
ix
Table of Contents
List of Tables ......................................................................................................... xi
List of Figures ....................................................................................................... xii
CHAPTER 1: Introduction ......................................................................................1
1.1 Statement of the Problem .......................................................................2
1.2 Research Objectives ...............................................................................3
1.3 Literature Review...................................................................................6
1.4 Outlines of This Dissertation ...............................................................13
CHAPTER 2: Understanding Fracture Initiation and Propagation Pressures .......16
2.1 Introduction ..........................................................................................17
2.2 Lost Circulation “Thresholds” .............................................................21
2.3 Fracture Initiation Pressure (FIP).........................................................22
2.4 Fracture Propagation Pressure (FPP) ...................................................30
2.5 Preventive and Remedial Wellbore Strengthening ..............................46
2.6 Summary ..............................................................................................50
CHAPTER 3: Developing a Framework for Lost Circulation Simulation ............53
3.1 Introduction ..........................................................................................54
3.2 Numerical Method and Governing Equations .....................................55
3.3 Lost Circulation Model ........................................................................64
3.4 Fluid Loss Simulation Results .............................................................70
3.5 Summary ..............................................................................................88
CHAPTER 4: Modeling Study of Preventive Wellbore Strengthening Treatments:
The Role of Mudcake....................................................................................90
4.1 Introduction ..........................................................................................91
4.2 An Analytical Mudcake Model ............................................................92
4.3 A Numerical Model for Time-dependent Mudcake ...........................112
4.4 Summary ............................................................................................140
x
CHAPTER 5: Modeling Study of Remedial Wellbore Strengthening Treatments142
5.1 Introduction ........................................................................................143
5.2 A Fracture-Mechanics-Based Analytical Model for Remedial Wellbore
Strengthening Applications ................................................................147
5.3 A Finite-element Model for Remedial Wellbore Strengthening
Applications .......................................................................................164
5.4 Summary ............................................................................................192
CHAPTER 6: Cement Interface Fracturing .........................................................193
6.1 Introduction ........................................................................................194
6.2 Development of Cement Interface Fracture Model ...........................197
6.3 Results and Discussion ......................................................................205
6.4 Summary ............................................................................................213
CHAPTER 7: Role of Field Injectivity Tests on Combating Lost Circulation ...214
7.1 Introduction ........................................................................................215
7.2 A Review of Filed Injectivity Tests ...................................................216
7.3 Test Signatures ...................................................................................223
7.4 Test Interpretation ..............................................................................228
7.5 Field Examples...................................................................................235
7.6 Developing a Simulation Framework for Injectivity Tests................239
7.7 Lost Circulation as a Function of Formation Lithology ....................255
7.8 Summary ............................................................................................259
CHAPTER 8: Conclusions and Future Work ......................................................261
8.1 Conclusions ........................................................................................262
8.2 Future Work .......................................................................................268
References ............................................................................................................270
xi
List of Tables
Table 3.1: Material properties of the static fluid loss model. ................................67
Table 3.2: Input parameters for the dynamic fluid loss model. .............................70
Table 4.1: Summary of the parameters used in the example cases. .....................104
Table 4.2: Input boundary-condition values for the mudcake model. .................117
Table 5.1: Input parameters for model validation. ...............................................156
Table 5.2: Base input parameters used in the proposed model. ...........................157
Table 5.3: Input parameters for the finite-element model. ..................................169
Table 6.1: Cement properties. ..............................................................................202
Table 6.2: Formation properties. ..........................................................................202
Table 6.3: Interface bond properties. ...................................................................202
Table 6.4: In-situ stresses, pore pressure and wellbore pressure applied to the model.
.........................................................................................................203
Table 6.5: Geometry of the initial cracks at the casing shoe. ..............................204
Table 7.1: Material properties for the simulations. ..............................................245
xii
List of Figures
Figure 2.1: Left: Pore pressure and fracture gradient plot in depleted zone. Pore
pressure decrease leads to a decrease in fracture gradient. Right: Pore
pressure and fracture gradient plot in deep-water formation with
abnormally high pressure. There is a reduced mud-weight window.18
Figure 2.2: Fracture-initiation pressure of a vertical well. Left: with different
horizontal stress anisotropies and pore pressure; Right: with different
𝛈 and pore pressure. .........................................................................25
Figure 2.3: FIP (micro-fracture-propagation pressure) decreases dramatically with an
increase in horizontal stress anisotropy, and increases moderately with
an increase in fracture toughness; it can be much smaller than the
minimum horizontal stress with high stress anisotropy and low fracture
toughness...........................................................................................27
Figure 2.4: Schematic pressure-volume/time curves in leak-off tests. Left: no visible
leak-off response at fracture initiation, the leak-off pressure is very close
to formation breakdown pressure; Middle: a clear leak-off point before
formation breakdown; Right: multiple leak-off points before formation
breakdown. ........................................................................................30
Figure 2.5: A small fracture on the wellbore wall before formation breakdown: Left:
no filter cake plugging with clean fluid; Middle: high solids
concentration or filter cake inside the fracture; Right: fracture is plugged
at the inlet on wellbore wall. .............................................................33
Figure 2.6: A large hydraulic fracture (wellbore at the fracture center is neglected).
...........................................................................................................35
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Figure 2.7: Pressure response during fracture propagation. Left: theoretical result;
Right: DEA-13 lab test result. ...........................................................36
Figure 2.8: A fracture plugged by solid particles. .................................................38
Figure 2.9: FIP with non-penetration zone length, pore pressure and minimum
horizontal stress. ...............................................................................39
Figure 2.10: A stationary fracture model. ...........................................................40
Figure 2.11: Fracture volume (left) and stress intensity factor (right) changes with
applying pore pressure to the fracture face only, while keeping the
traction on the fracture face constant. ...............................................41
Figure 2.12: Hydraulic fractures in impermeable (left) and permeable (right)
formations with high solids content fluid. ........................................42
Figure 2.13: Fluid leak-off and filter plug development controlled by capillary
pressure for water-wet sandstone with larger pore size and shale with
smaller pore size. Formation fluid is water while fracture fluid is
oil/synthetic based mud.....................................................................44
Figure 2.14: Fluid leak-off and filter cake development controlled by fluid
immiscibility and capillary pressure. ................................................45
Figure 2.15: An example of a preventive wellbore strengthening test on a sandstone
block (after Guo et al., 2014). ...........................................................48
Figure 2.16: The sandstone test block for pressure build-up curve in Figure 12. The
block was fractured to the edges without obvious fluid leak-off due to
LCM sealing effect (after Guo et al., 2014). .....................................49
xiv
Figure 2.17: A repeated hydraulic fracturing test with LCM. First injection cycle
(preventive treatment - intact wellbore, with LCM): high leak-off
pressure and high propagation pressure. Second injection cycle
(fractured wellbore, without LCM): low leak-off pressure and low
propagation pressure. Third injection cycle (fractured wellbore, with
LCM): low leak-off pressure, high (increased) propagation pressure.
(after Black et al., 1988). ..................................................................49
Figure 2.18: Preventive wellbore strengthening treatment enhances both leak-off
pressure and FPP, while remedial wellbore strengthening treatment only
enhances FPP. ...................................................................................50
Figure 3.1: Illustration of lost circulation system with well, formation and fracture.
...........................................................................................................56
Figure 3.2: Schematic of fluid flow in pipe. ..........................................................58
Figure 3.3: Friction factor determined from Blasius model and Churchill model:
Blasius model shows discontinuous transition from laminar to turbulent
flow at Re=2500, while Churchill model shows smooth transition. .59
Figure 3.4: A typical traction-separation law ........................................................61
Figure 3.5: Schematic of fluid flow in the cohesive fracture (Modified after Zielonka
et al., 2014) .......................................................................................62
Figure 3.6: Schematic configuration of well and formation. The formation is in a
plane-strain condition........................................................................65
Figure 3.7: Static lost circulation model. ...............................................................66
Figure 3.8: Dynamic fluid loss model....................................................................68
Figure 3.9: Fluid loss rate with different mud densities in the static fluid loss model.
...........................................................................................................72
xv
Figure 3.10: BHP with different mud densities in the static fluid loss model. ......72
Figure 3.11: Lost circulation fracture geometry with different mud densities in the
static fluid loss model. ......................................................................73
Figure 3.12: Return circulation rate with different mud densities in the dynamic fluid
loss model. ........................................................................................75
Figure 3.13: Fluid loss rate with different mud densities in the dynamic fluid loss
model.................................................................................................76
Figure 3.14: BHP with different mud densities in the dynamic fluid loss model. .76
Figure 3.15: Lost circulation fracture geometry with different mud densities in the
dynamic fluid loss model. .................................................................77
Figure 3.16: Comparison of fracture geometry between the static and dynamic fluid
loss models. .......................................................................................78
Figure 3.17: Return circulation rate with different mud viscosities in the dynamic
fluid loss model. ................................................................................79
Figure 3.18: Fluid loss rate with different mud viscosities in the dynamic fluid loss
model.................................................................................................80
Figure 3.19: BHP with different mud viscosities in the dynamic fluid loss model.80
Figure 3.20: Lost circulation fracture geometry with different mud viscosities in the
dynamic fluid loss model. .................................................................81
Figure 3.21: Return circulation rate with different pump rates in the dynamic fluid
loss model. ........................................................................................83
Figure 3.22: Fluid loss rate with different pump rates in the dynamic fluid loss model.
...........................................................................................................83
Figure 3.23: BHP with different pump rates in the dynamic fluid loss model. .....84
xvi
Figure 3.24: Lost circulation fracture geometry with different pump rates in the
dynamic fluid loss model. .................................................................85
Figure 3.25: Return circulation rate with different annulus clearances in the dynamic
fluid loss model. ................................................................................86
Figure 3.26: Fluid loss rate with different annulus clearances in the dynamic fluid loss
model.................................................................................................87
Figure 3.27: BHP with different annulus clearances in the dynamic fluid loss model.
...........................................................................................................87
Figure 3.28: Lost circulation fracture geometry with different annulus clearances in
the dynamic fluid loss model. ...........................................................88
Figure 4.1: Schematic of the cross section of wellbore, mudcake, and formation.93
Figure 4.2: Pore Pressure distribution around wellbore with different mudcake
thickness 𝑤. ....................................................................................105
Figure 4.3: Total tangential stress induced by fluid flow with different mudcake
thickness 𝑤. ....................................................................................106
Figure 4.4: Effective tangential stress induced by fluid flow (compared with no flow
case) with different mudcake thickness 𝑤. ....................................106
Figure 4.5 Fracture pressure with different mudcake thickness 𝑤. ....................107
Figure 4.6: Pore Pressure distribution around wellbore with different mudcake
permeability 𝐾1. ............................................................................109
Figure 4.7: Total tangential stress induced by fluid flow with different mudcake
permeability 𝐾1. ............................................................................109
Figure 4.8: Effective tangential stress induced by fluid flow (compared with no flow
case) with different mudcake permeability 𝐾1. .............................110
Figure 4.9: Fracture pressure with different mudcake permeability 𝐾1. ............110
xvii
Figure 4.10: Effective tangential stress around wellbore with different mudcake yield
strength 𝑌. ......................................................................................111
Figure 4.11: Fracture pressure with different mudcake yield strength 𝑌. ...........112
Figure 4.12: Illustration of mudcake model with constant mudcake thickness 𝑤𝑜 and
equivalent mudcake permeability 𝑘𝑒𝑡. ..........................................114
Figure 4.13: Geometry of the finite-element mudcake model. ............................115
Figure 4.14: Boundary conditions of the finite-element mudcake model. ..........116
Figure 4.15: Variation of mudcake thickness with time. .....................................119
Figure 4.16: Variation of mudcake permeability with time. ................................119
Figure 4.17: Variation of equivalent mudcake permeability with time. ..............120
Figure 4.18: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for
wellbore with extremely permeable mudcake and permeable wellbore
without mudcake. ............................................................................122
Figure 4.19: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3
hours for wellbore with extremely permeable mudcake and permeable
wellbore without mudcake. .............................................................122
Figure 4.20: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for
wellbore with extremely permeable mudcake and permeable wellbore
without mudcake. ............................................................................123
Figure 4.21: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for
wellbore with impermeable mudcake and impermeable wellbore without
mudcake. .........................................................................................123
Figure 4.22: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3
hours for wellbore with impermeable mudcake and impermeable
wellbore without mudcake. .............................................................124
xviii
Figure 4.23: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for
wellbore with impermeable mudcake and impermeable wellbore without
mudcake. .........................................................................................124
Figure 4.24: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes. 126
Figure 4.25: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours. ......126
Figure 4.26: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30
minutes. ...........................................................................................127
Figure 4.27: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
.........................................................................................................127
Figure 4.28: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30
minutes. ...........................................................................................128
Figure 4.29: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4
hours. ...............................................................................................128
Figure 4.30: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes. 130
Figure 4.31: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours. ......130
Figure 4.32: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30
minutes. ...........................................................................................131
Figure 4.33: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
.........................................................................................................131
Figure 4.34: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30
minutes. ...........................................................................................132
Figure 4.35: Effective tangential stress profiles along SH direction at t = 4 hours.132
Figure 4.36: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes. 134
Figure 4.37: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours. ......135
xix
Figure 4.38: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30
minutes. ...........................................................................................135
Figure 4.39: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
.........................................................................................................136
Figure 4.40: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30
minutes. ...........................................................................................136
Figure 4.41: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4
hours. ...............................................................................................137
Figure 4.42: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes. 139
Figure 4.43: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30
minutes. ...........................................................................................139
Figure 4.44: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30
minutes. ...........................................................................................140
Figure 5.1: Comparison of fracture pressure with and without remedial wellbore
strengthening treatment in DEA 13 experimental study (reproduced from
Onyia, 1994). ..................................................................................144
Figure 5.2: Results of a laboratory wellbore strengthening test (reproduced from Guo
et al., 2014). ....................................................................................145
Figure 5.3: Results of LOT field tests before and after taking remedial wellbore
strengthening treatment (reproduced from Aston et al., 2004). ......146
Figure 5.4: Schematic of remedial wellbore strengthening problem. ..................149
Figure 5.5: Problem decomposition. Left: an intact wellbore subject to wellbore
pressure and anisotropic far-field horizontal stresses; Right: wellbore
with two symmetric fractures subject to fluid pressure on fracture
surfaces. ..........................................................................................150
xx
Figure 5.6: Fracture in an elastic solid subject to a couple of point loads. ..........151
Figure 5.7: Wellbore with two symmetric fractures subject to uniform pressure on the
wellbore and fracture surfaces and anisotropic far-field horizontal
stresses. ...........................................................................................155
Figure 5.8: Comparison of fracture tip stress intensity factors between the proposed
model and Tada’s model (Tada et al., 1985). .................................156
Figure 5.9: Fracture-tip stress intensity factor with different horizontal stress
anisotropies and bridge locations. ...................................................159
Figure 5.10: Fracture pressure with different horizontal stress anisotropies and bridge
locations. .........................................................................................159
Figure 5.11: Fracture-tip stress intensity factor with different pore pressure and bridge
locations. .........................................................................................161
Figure 5.12: Fracture pressure with different pore pressure and bridge locations.161
Figure 5.13: Fracture pressure with different fracture toughness of the formation rock.
.........................................................................................................162
Figure 5.14: The remedial wellbore strengthening model. Left: geometry and
boundary conditions of the model; Rigth: detailed fracture process zone.
.........................................................................................................167
Figure 5.15: Comparison of stress intensity factor for unbridged fracture. .........171
Figure 5.16: Comparison of stress intensity factor for bridged fracture with various
bridging locations............................................................................172
Figure 5.17: Hoop stress distribution before (left) and after (right) bridging the
fracture in remedial wellbore strengthening treatment. ..................173
xxi
Figure 5.18: Hoop stress on wellbore wall for different horizontal stress contrasts: (a)
without fracture, (b) with unbridged fracture, (c) with bridged fracture,
and (d) comparison of hoop stresses before and after bridging for
horizontal stress contrast equal to 1.3. ............................................175
Figure 5.19: Hoop stress on wellbore wall for different bridging locations. .......176
Figure 5.20: Hoop stress on wellbore wall for different leak-off rates: (a) without
fracture, (b) with unbridged fracture, (c) with bridged fracture, and (d)
comparison of hoop stresses before and after bridging the fracture for
leak-off rate equal to 4 inches/s. .....................................................177
Figure 5.21: Hoop stress along fracture face for different horizontal stress contrasts:
(a) without fracture, (b) with unbridged fracture, (c) with bridged
fracture, and (d) comparison of hoop stresses before and after bridging
the fracture for horizontal stress contrast equal to 1.3. ...................179
Figure 5.22: Hoop stress along fracture face for different bridge locations. .......180
Figure 5.23: Hoop stress along fracture face for different leak-off rates: (a) without
fracture, (b) with unbridged fracture, (c) with bridged fracture, and (d)
comparison of hoop stresses before and after bridging the fracture for
leak-off rate equal to 4 inches/s. .....................................................182
Figure 5.24: Hoop stress along fracture face for different pressures behind LCM
bridge, varying from formation pressure of 1800 psi to wellbore pressure
of 4000 psi.......................................................................................183
Figure 5.25: Vertical displacement distribution in the model before and after bridging
the fracture in wellbore strengthening. ...........................................185
xxii
Figure 5.26: Fracture half-width distribution for different horizontal stress contrasts:
a) before bridging the fracture, b) after bridging the fracture, and c)
comparison of fracture half-widths before and after bridging the fracture
for horizontal stress contrast equal to 1.3. ......................................186
Figure 5.27: Fracture half-width distribution for different LCM bridge locations.186
Figure 5.28: Fracture half-width distribution for different leak-off rates: (a) before
bridging the fracture, (b) after bridging the fracture. ......................187
Figure 5.29: Fracture half-width distribution for different Young’s Modulus before
and after bridging the fracture.........................................................188
Figure 5.30: Fracture half-width distribution for different Poisson’s ratios before and
after bridging the fracture. ..............................................................189
Figure 5.31: Fracture half-width distribution for different pressure behind LCM
bridge, varying from formation pressure 1800 psi to wellbore pressure
4000 psi. ..........................................................................................190
Figure 6.1: Cement sheath model. (a) the one-quarter geometry; (b) top view of the
casing/cement/formation system; (c) the interface between casing and
cement. ............................................................................................200
Figure 6.2: Interface fracture opening of two cases with uniform horizontal stress
(SR1) and non-uniform horizontal stress (SR1.2). The pictures are top
views of the cut sections of the casing/cement/interface system at 0.5 m
above the casing shoe......................................................................206
Figure 6.3: Interface fracture width around wellbore for the two cases in Figure 6.2.
0-degree and 90-degree correspond to the directions of the maximum
and minimum horizontal stress, respectively. .................................207
xxiii
Figure 6.4: Developments of fracture geometry, fracture pressure and cement plastic
strain with different initial crack sizes. The pictures are front views of
the one-quarter model, with maximum horizontal stress in the X-axis
direction and minimum horizontal stress in the Y-axis direction. The
circumferential extents of the initial cracks for cases Q1 through Q4 are
30𝑜, 45𝑜, 60𝑜 and 90𝑜, respectively, as given in Table 5.5. ....209
Figure 6.5: Developments of fracture geometry and fracture pressure with different
cement properties. The pictures are front views of the one-quarter
model, with maximum horizontal stress in the X-axis direction and
minimum horizontal stress in the Y-axis direction. The properties for
cement types C1 through C3 are given in Table 6.1. ......................211
Figure 6.6: Development of fracture geometry and fracture pressure with different
formation properties. The pictures are front views of the one-quarter
model, with maximum horizontal stress in the X-axis direction and
minimum horizontal stress in the Y-axis direction. The properties for
formation types F1 through F3 are given in Table 2. .....................212
Figure 7.1: Schematic illustration of pressure-time/volume plot in a leak-off test.218
Figure 7.2: Near wellbore hoop stress concentration with uniform far-field stresses
𝑆ℎ𝑚𝑖𝑛 = 𝑆𝐻𝑚𝑎𝑥 (blue color = more compression, red color = less
compression). ..................................................................................218
Figure 7.3: Typical XLOT plot (Modified after Gaarenstroom et al., 1993)....221
Figure 7.4: An example of pressure-volume plot in a pump-in and flow-back test
(Modified after Gederaas and Raaen, 2009). ..................................222
Figure 7.5: Pressure-time plot of a typical PIFB test. .......................................225
xxiv
Figure 7.6: “Saw-tooth” pressure response during fracture propagation (Reproduced
from Okland et al., 2002). ...............................................................225
Figure 7.7: Pressure-volume plot of a typical PIFB test (Modified after Fjar et al.,
2008). ..............................................................................................227
Figure 7.8: Total volume pumped into the well before fracture creation (Modified
after Altun, 1999 and Fu, 2014). .....................................................227
Figure 7.9: FIP=LOP=FBP in idealized condition. ..........................................228
Figure 7.10: FIP, LOP and FBP are not identical with drilling mud as injection
fluid. ................................................................................................230
Figure 7.11: Permeable formation has a large pressure decline during shut-in due to
sufficient leak-off from fracture, while a relative flat pressure response is
usually observed in an impermeable formation. .............................234
Figure 7.12: An example of predicting FCP using flow-back data (Reproduced from
Gederaas and Raaen, 2009). ............................................................235
Figure 7.13: Test in Well 11-2 in formation with relatively high permeability
(Reproduced from Okland et al., 2002). .........................................236
Figure 7.14: Test in Well 10-7 in formation with relatively low permeability
(Reproduced from Okland et al., 2002). .........................................236
Figure 7.15: General PIFB test system with well, formation and fracture. .........242
Figure 7.16: PIFB test model. Top: geometry and boundary conditions of the model;
Bottom: refined mesh around the wellbore. ....................................244
Figure 7.17: Bottom hole pressure versus time plots. Top: PIFB test in permeable
formation; Bottom: PIFB test in impermeable formation. ..............249
Figure 7.18: Pressure versus square root of time plot during shut-in. Top: Permeable
formation; Bottom: Impermeable formation. ..................................251
xxv
Figure 7.19: BHP vs time during flowback phases in impermeable formation.254
Figure 7.20: BHP versus added fluid volume in the system during shut-in and
flowback phases in the impermeable test. .......................................254
Figure 7.21: BHP versus added fluid volume in the system during shut-in and
flowback phases in the permeable test. ...........................................255
Figure 7.22: Two lost circulation and wellbore breathing events occurred in silty
shale formations, rather than in clean shale or clean sand formations.259
1
CHAPTER 1: Introduction
In this dissertation, a systematic study on the lost circulation and wellbore
strengthening is performed. This chapter introduces the background of the problem, the
objectives of this dissertation, a literature review on lost circulation and wellbore
strengthening, and the organization of this dissertation.
2
1.1 STATEMENT OF THE PROBLEM
Lost circulation is the loss of partial or whole drilling fluid into the rock formation
during drilling a well. It is a major contribution to non-productive time (NPT) in drilling
industry (Cook et al., 2011). Lost circulation can lead to issues such as differential sticking
and well control events which can further increase NPT and drilling cost (Shahri et al.,
2014).
More than 12% of NPT has been reported for Gulf of Mexico area shelf drilling
due to lost circulation alone (Wang et al., 2007a). The US Department of Energy reported
that on average 10% to 20% of the cost of drilling high-pressure and high-temperature
(HTHP) wells is expended on mud losses (Growcock et al., 2009). The impact of lost
circulation on well construction is significant, representing an estimate 2 to 4 billion dollars
annually in lost time, lost drilling fluid and materials used to stem the losses (Cook et al.,
2011).
Most of the lost circulations events are fracture initiation and propagation problems,
occurring when the fluid pressure in the wellbore is high enough to create fractures on the
wellbore wall. Lost circulation usually happens in formations with narrow drilling mud
weight window between pore pressure/collapse pressure gradient and fracture gradient.
Three typical scenarios are fluid losses in depleted zones, deepwater formations and
deviated wellbores. In depleted zones, pore pressure reduction usually leads to a significant
decrease in fracture gradient; so it’s much easier to fracture the wellbore during drilling. In
deepwater formations, the fracture gradient is relative low and the mud weight window is
relative narrow since sea water cannot provide as much as overburden loading as rock does;
therefore it’s a challenge to maintain wellbore pressure in this narrow window considering
of friction pressure and swab/surge pressure during drilling. In deviated wellbores, with
the increase of borehole inclination, the mud weight window can diminish very quickly,
3
even resulting in a zero drilling margin; so wellbore pressure is very likely to overcome
fracture gradient, leading to a lost circulation event.
In order to drilling through problematic zones with high risk of lost circulation,
drilling engineers use some approaches to artificially increase the fracture pressure
(maximum pressure a wellbore can sustain without significant fluid loss) and hence widen
drilling mud weight window by bridging, plugging or sealing the fractures. These
approaches are called “Wellbore Strengthening”. Currently, there are mainly two types of
wellbore strengthening methods in drilling industry. They are hoop stress enhancement
method, e.g. Stress Cage (Alberty and McLean, 2004) and Fracture Closure Stress
(Dupriest, 2005), and fracture resistance enhancement method, e.g. Fracture Propagation
Resistance (van Oort et al., 2011) and Tip Screen-out (Fuh et al., 1992; Morita et al., 1990).
Although a few successful applications of wellbore strengthening have been
reported, there is still a lack of a clear understanding of its fundamentals and a lot of
disagreements still exist. Therefore, a compressive investigation of fracture behaviors
during lost circulation and wellbore strengthening is necessary and crucial for
understanding their fundamentals and designing proper wellbore strengthening treatments.
In this dissertation, an integrated research on lost circulation and wellbore strengthening is
conducted based on both analytical and numerical studies.
1.2 RESEARCH OBJECTIVES
The overall objective of this dissertation is to study fracture behaviors during lost
circulation and wellbore strengthening and provide useful implication for lost circulation
prevention/remediation. The research objective will be achieved by carrying out the
following tasks.
4
1.2.1 Understanding Fracture Initiation and Propagation Pressures
The first objective of this dissertation is to perform a detailed investigation on
fracture initiation and propagation pressures for understanding lost circulation and
wellbore strengthening. For significant fluid loss to occur, fractures must initiate on an
intact wellbore or reopen on a wellbore with preexisting fractures, and then propagate into
far-field region. Wellbore strengthening operations are designed to increase one or both of
these two pressures in order to inhibit fracture growth. Some theoretical models used in
drilling industry assume fracture initiation and propagation pressures are only functions of
in-situ stress and rock mechanical properties. However, as demonstrated by numerous field
and laboratory observations, they are also highly related to drilling fluid properties, and
interactions between drilling fluid and formation rock. A detailed discussion on factors that
can affect fracture initiation and propagation pressures will be first presented in this
dissertation. These factors include: micro-fractures on the wellbore wall, in-situ stress
anisotropy, pore pressure, fracture toughness, filter cake development, fracture
bridging/plugging, bridge location, fluid leak-off, rock permeability, pore size of rock, mud
type, mud solid concentration, and critical capillary pressure.
1.2.2 Numerical Investigation of Lost Circulation
The second objective of this dissertation is to develop a coupled fluid flow and
geomechanics model to simulate lost circulation while drilling. The numerical model
should take into account the main elements during fluid loss, including drilling fluid
circulation, fracture growth, fracture fluid flow, pore fluid flow, and deformation of
formation rock. The model should be able to not only capture the essential signatures of
lost circulation known from field observations such as the reduction in return circulation
rate, but also predict those we cannot measure during drilling such as the development of
fracture geometry with fluid loss.
5
1.2.3 Understanding the Role of Mudcake in Preventive Wellbore Strengthening
Treatments
The third objective of this research is to develop models to investigate the
preventive wellbore strengthening method based on plastering wellbore surface with
mudcake. First an analytical model assuming steady-state fluid flow will be developed to
investigate the effects of mudcake thickness, permeability and strength on stress
distribution and fracture pressure of a wellbore. Second, a numerical model will be
developed to exam the transient effects of dynamic mudcake thickness buildup coupled
with dynamic mudcake permeability reduction on the evolution of pore pressure and stress
around wellbore during drilling.
1.2.4 Modeling Studies of Remedial Wellbore Strengthening Treatments
The fourth objective of this research is to develop models to investigate the
remedial wellbore strengthening method based on plugging the fractures using lost
circulation materials (LCMs). First, an analytical solution based on linear elastic fracture
mechanics will be proposed for investigating the geomechanical aspects of wellbore
strengthening operations. The proposed solution should be able to take into account the
effects of wellbore-fracture geometry, in-situ stress anisotropy, and LCM bridge location
on wellbore strengthening, and provide a fast procedure to predict fracture pressure change
before and after bridging a fracture. Second, a finite-element model will be developed for
obtaining the detailed stress and deformation information in remedial wellbore
strengthening treatment, with a focusing attention on the evolutions of near-wellbore hoop
stress and fracture width.
1.2.5 Investigating Fluid Loss through Casing Shoe
The fifth objection is to develop a coupled fluid flow and geomechanics model to
simulate fluid leakage through casing shoe and cement interface. A three-dimensional
6
finite-element model will be developed to simulate cement interface debonding and fluid
flow along the debonding fracture due to pressure buildup at the casing shoe.
1.2.6 Revealing the Importance of Pump-in and Flow-back Tests for Combatting
Lost Circulation
The sixth objective of this dissertation is to reveal the importance of pump-in and
flow-back tests for combatting lost circulation. Striking similarities exist between the
pump-in and flow-back test and lost circulation, since both of them are issues of fracture
growth from wellbore wall to far-field region. A detailed phase by phase interpretation of
pump-in and flow-back tests will be conducted for helping understand fracture behaviors
(initiation, propagation and closure) during lost circulation. A finite-element framework
for modeling field injectivity tests with fluid injection, well shut-in and fluid flowback will
also be developed. Several methods for estimating minimum principal stress using shut-in
data and flow-back data will be investigated.
1.3 LITERATURE REVIEW
1.3.1 Experimental Studies of Lost Circulation
Even though lost circulation is a major NPT event in drilling industry, very few
experimental studies have been conducted on it. The joint-industry DEA-13 experimental
study conducted in the middle 1980’s to early 1990’s (Fuh et al., 1992; Morita et al., 1996a,
1990, 1990; Onyia, 1994) is an early experimental investigation into lost circulation. The
aim of this study is to determine why lost circulation occurs less frequently while drilling
with water based mud (WBM) than with oil based mud (OBM), or why OBM apparently
causes a lower fracture gradient than WBM. Most of the DEA-13 experiments were
conducted using 30×30×30 inch sandstone blocks with 1.5-inch boreholes. Two major
observations of DEA-13 project are: (1) fracture initiation pressure (FIP) and fracture
7
reopening pressure (FRP) are almost independent of mud type and use of LCM additives,
and (2) fracture propagation pressure (FPP) was found to be strongly related to mud type
and significantly increased by use of LCM additives. The experimental results were
explained by a physical model called “tip screen-out” (Fuh et al., 1992; Morita et al., 1996a,
1996b, 1990; Morita and Fuh, 2012), which indicates that fracture gradient (i.e. FPP)
increases due to a sealing plug near fracture tip which isolates fracture tip from wellbore
pressure.
Another experimental effort to study lost circulation is the GPRI 2000 joint-
industry project conducted in the late 1990’s to early 2000’s ( Dudley et al., 2001; van Oort
et al., 2011). Different from DEA-13 experiments conducted on relative large rock blocks,
thus allowing the researchers to observe fracture propagation, GPRI 2000 experiments
were mostly performed with 4-inch diameter cores with 5/8-inch boreholes. Only FIP and
FRP can be evaluated, while FPP cannot be observed with such small samples. The purpose
of GPRI 2000 study was to evaluate the capabilities of different LCMs on increasing
fracture pressure, with a focus on increasing FRP. It was found that when LCMs were used
in the mud, FRP can be significantly increased, which was contrary to the observation in
DEA-13 study; and LCM was more effective in increasing FRP in WBM than in OBM or
synthetic based mud (SBM).
A recent experimental study project on lost circulation was conducted in M-I
SWACO from late 2000’s to early 2010’s (Guo et al., 2014), which was called Lost
Circulation and Wellbore Strengthening Research Cooperative Agreement (RCA) project.
RCA project has been conducted to investigate wellbore strengthening mechanism and
effectiveness of various wellbore strengthening methods, including preventive
strengthening method and remedial strengthening method. Both sandstone and shale blocks
of 6×6×6 inch with borehole of 1 inch were used in the tests. The results show that: (1) a
8
preventive wellbore strengthening treatment is more effective than remedial treatment; (2)
particle size distribution (PSD) and concentration of LCM are critical in sealing/bridging
the fractures; and (3) fracture pressure achieved with wellbore strengthening is always
higher than the breakdown pressure, which means wellbore strengthening can not only
repair weak formation with induced/natural fractures, but also strengthen the formation.
1.3.2 Physical Models for Wellbore Strengthening
Different from its literal meaning, “Wellbore Strengthening” does not target on
increasing the strength of wellbore or formation. It is an operation to alter the stress
distribution in the vicinity of wellbore and fracture and/or fluid pressure distribution inside
the fracture to increase the fracture pressure (i.e. the maximum sustainable pressure of a
wellbore without significant fluid loss). Many field applications and laboratory tests have
shown that fracture pressure can be significantly increased by wellbore strengthening
operations (Alberty and McLean, 2004; Aston et al., 2007, 2004a; Duffadar et al., 2013;
Dupriest, 2005; Dupriest et al., 2008; Guo et al., 2014; van Oort et al., 2011; van Oort and
Razavi, 2014; Wang et al., 2009, 2007a, 2007a, 2007b). Currently, there are three major
physical models in the drilling industry for explaining why wellbore strengthening
treatments can “strengthen” a wellbore.
Stress Cage (SC) model (Alberty and McLean, 2004). When a fracture is created
on wellbore wall, LCM particles are forced in to the fracture. The largest particles first
wedge the fracture mouth on the wellbore wall. Then, smaller LCM particles come in and
plug the spaces among larger particles and between particles and fracture surfaces, and
then seal the fracture mouth together with filtration control agents. Next, trapped fluid in
the fracture filters into the formation through fracture surfaces and compressive forces are
transferred to LCM bridge at fracture mouth. Finally, the fracture is bridged at fracture
9
mouth resulting in an increased hoop stress near the bridge location, which makes the
fracture more difficult to open.
Fracture Closure Stress (FCS) model (Dupriest, 2005). Fractures on wellbore
wall are first created and widened to increase the compressive stress, i.e. fracture closure
stress, in the adjacent rock. The greater the fracture width, the greater the fracture closure
stress. Next, LCM particles within mud slurry are forced into the fracture. Liquid leaks off
from the slurry to the formation rock. LCM particles consolidate and finally form a bridge
inside the fracture that keeps fracture open and isolates fracture tip from wellbore pressure.
The increased fracture closure stress and isolation of fracture tip make the fracture more
difficult to open and extend.
Fracture Propagation Resistance (FPR) model (Morita et al., 1996b, 1990; van
Oort et al., 2011). FPR model does not aim to alter the near wellbore stress to increase hoop
stress or fracture closure stress, instead it attempts to increase the formation’s resistance
against fracture propagation. FPR model supposes that a filter cake can form inside the
fracture as fracture propagation. The filter cake can seal fracture tip and prevent pressure
communication between fracture tip and wellbore, therefore the resistance for fracture
propagation can be increased. FPR model argues that fracture initiation and reopen pressure
cannot be increased by wellbore strengthening treatments, but rather fracture propagation
pressure can be significantly increased.
1.3.3 Analytical Studies of Lost Circulation and Wellbore Strengthening
Compared to other drilling problems, e.g. wellbore instability, there are very few
analytical studies for lost circulation and wellbore strengthening. Several parameters that
researchers want to know from analytical studies are fracture pressure, fracture width and
fracture-tip stress intensity factor before and after bridging/sealing the fracture. For
10
calculating these parameters, it is important to consider the near wellbore stress
concentration for short fracture and pressure drop along fracture for large fracture.
Based on a dimensional analysis and superposition principle of linear elastic
fracture mechanics, Guo et al. (2011) gave an approximate, closed-form solution for the
crack mouth opening displacement (CMOD) of two fractures symmetrically located at
wellbore wall with the fracture surface subjected to either uniform wellbore pressure or
pore pressure. Their model is only valid for a relative short fracture with a length less than
4 wellbore radii and for the maximum to minimum horizontal stress ratio less than 2.
Shahri et al. (2014) proposed a semi-analytical solution for wellbore strengthening
analysis based on singular integral formulation of stress field and solved using Gauss-
Chebyshev polynomials. Their model considers far field stress anisotropy and near
wellbore stress perturbation, but doesn’t take into account pressure drop along fracture and
can’t give closed-form solutions for fracture width and fracture pressure.
Ito et al. (2001) applied the penny-shaped hydraulic fracture model developed by
Abé et al. (1976) to analyze fracture pressure increase after plugging the fracture. However,
there model does not take into account near wellbore stress concentration (i.e. the existence
of wellbore is neglected) and therefore can only be used for a fracture with a size much
larger than wellbore size. What’s more, this model assumes the pressure is uniform inside
the fracture from wellbore to LCM plug and does not consider pressure drop, which is not
realistic for a large fracture.
Using linear elastic fracture mechanics and superposition principle, Morita and Fuh
(2012) proposed two sets of closed-form solutions for stress intensity factor and fracture
pressure after bridging the fracture at fracture mouth and away from fracture mouth. They
assumed the pressure inside the fracture from fracture mouth to bridge location was
uniform and equal to wellbore pressure. Their model for bridging fracture away from
11
fracture mouth neglects the effect of wellbore on stress intensity factor induced by pressure
from LCM bridge to fracture tip, therefore it is only valid for large fracture for which it is
reasonable to ignore the effect of wellbore.
van Oort and Razavi (2014) extended the KGD hydraulic fracture model (Geertsma
and De Klerk, 1969; Linkov, 2013; Zheltov, 1955) to analyze wellbore strengthening.
Fracture pressure and fracture width after sealing the fracture is derived, but this model
neglects both wellbore effect (i.e. near wellbore stress concentration) and pressure drop in
the fracture.
1.3.4 Numerical Studies of Lost Circulation and Wellbore Strengthening
Since, analytical solutions, that describe the fracture geometry and
stress/displacement field near wellbore and fracture and satisfy our needs for wellbore
strengthening analysis, are still not available in the literature (Wang et al., 2009, 2007a),
numerical methods can be applied to simulate the problem and provide valuable
implications for field application. But, so far, only few numerical studies have been
conducted for lost circulation and wellbore strengthening problems, even though there have
been numerous numerical studies for wellbore stability and hydraulic fracturing problems
which are closely related to lost circulation/wellbore strengthening.
Wang et al. (2009, 2007a) proposed a 2D boundary element model to simulate two
symmetric fractures on wellbore wall under anisotropic in-situ stresses and get the stress
and fracture width distribution before and after bridging the fracture in wellbore
strengthening. Guo et al. (2011) used a 2D finite element model to investigate the fracture
width distribution for two pre-existing fractures symmetrically located at the wellbore wall
under various in-situ stress and fracture length; but the simulation did not give any
information about fracture behavior after applying wellbore strengthening treatments.
12
Alberty and McLean (2004) employed a 2D finite-element model to study fracture width
distribution and hoop stress field after bridging the fracture near its mouth under nearly
isotropic in-situ stresses. All the above numerical models assume the rock is linearly elastic
and do not take into account the porous features of the rock, therefore the effect of fluid
flow inside the rock and fluid leak-off through wellbore and fracture surfaces are not
considered.
With the goal to perform a comprehensive parametric study for wellbore
strengthening, Arlanoglu et al. (2014), Feng et al. (2015a) and Feng and Gray (2016a)
developed a 2D finite element model assuming the rock is poroelastic material and
including fluid flow inside the rock and across fracture and wellbore surfaces. Based on
their model, stress and pore pressure fields and fracture width distribution before and after
bridging a fracture were investigated, and a comprehensive parametric sensitivity studies
for wellbore strengthening were performed.
With the aim to investigate the hypothesis of wellbore hoop stress
increases when fractures are wedged and/or sealed as presented in Stress Cage
theory, Salehi (2012) and Salehi and Nygaard (2011) used cohesive element method to
study fracture propagation and sealing during lost circulation and wellbore strengthening.
According to their simulation results, they argued that fracture sealing/bridging is not able
to increase wellbore hoop stress more than its ideal state when no fracture exists. But in
their model, a constant injection rate (fluid loss rate) boundary condition was defined at
the fracture inlet which is not consistent with the actual drilling situation where the
downhole condition at fracture inlet is neither a constant flow rate nor a constant pressure.
13
1.4 OUTLINES OF THIS DISSERTATION
This dissertation consists of eight chapters. This chapter (Chapter 1) provides a
brief statement of the problems of lost circulation and wellbore strengthening, an
introduction to the objectives of this dissertation, and a literature review on lost circulation
and wellbore strengthening.
Chapter 2 discusses the role of fracture initiation and propagation pressures on lost
circulation and wellbore strengthening. In view of the existing disagreements on the
fundamentals of lost circulation and wellbore strengthening, a critical and detailed analysis
of fracture initiation and propagation pressures is conducted. Factors that may affect these
two pressures are investigated, which include micro-fractures on the wellbore wall, in-situ
stress anisotropy, pore pressure, fracture toughness, filter cake development, fracture
bridging/plugging, bridge location, fluid leak-off, rock permeability, pore size of rock, mud
type, mud solid concentration, and critical capillary pressure.
Chapter 3 introduces a finite-element model to simulate lost circulation during
drilling with circulation of drilling fluid. Circulation flow of drilling fluid in the well “U-
Tube” consisting of drilling pipe and annulus is simulated based on Bernoulli’s theory.
Fracture propagation into the porous rock is modeled using coupled pore pressure cohesive
zone method. The two parts are coupled together to predict the dynamic fluid loss and
fracture geometry evolution in drilling process. The numerical model provides a unique
new way to model lost circulation while drilling when the boundary condition at the
fracture mouth is neither a constant flowrate nor a constant pressure, but rather a dynamic
bottom-hole pressure.
In Chapter 4, an analytical solution and a numerical model are developed to
investigate the role of mudcake on preventive wellbore strengthening treatments based on
plastering wellbore wall with mudcake. The analytical solution is derived based on steady-
14
state flow assumption and superposition principle. It incorporates the effects of thickness,
permeability and strength of mudcake on near-wellbore pore pressure and stress
distribution, hence the effectiveness of preventive wellbore strengthening treatment. In
order to describe the effects of dynamic mudcake buildup and time-dependent mudcake
property (permeability) on wellbore strengthening. A finite-element framework based on
poroealstic theory is developed to investigate the transient effects of mudcake thickness
buildup and mudcake permeability change on the near-wellbore stress and pore pressure,
and thus the strengthening of wellbore.
In Chapter 5, an analytical solution and a finite-element model are proposed for
modeling remedial wellbore strengthening treatment based on plugging/bridging the
fractures using LCMs. The analytical model, based on linear elastic fracture mechanics
theory, provides a fast procedure to predict fracture pressure change before and after
fracture bridging. The numerical model takes into account poromechanical effects and
provides a more accurate prediction of local stress distribution and fracture width with
wellbore strengthening operations. Sensitivity analyses are then performed using both of
the models to quantify the effects of rock properties, in-situ stresses, bridge locations and
fluid flow on remedial wellbore strengthening.
In Chapter 6, a three-dimensional finite-element framework is developed to exam
the possibility of fluid leakage through casing shoe and along the weak cement interface
when there is pressure buildup in the wellbore due to change of drilling or completion fluid,
conduction of injectivity tests, and etc.
Chapter 7 highlights the importance of field injectivity tests for understanding the
fundamentals of lost circulation and wellbore strengthening, with a review of different
kinds of field tests and a discussion of their advantage and limitations. A coupled fluid flow
and geomechanics injectivity model is also developed which can capture the key elements
15
of injectivity tests known from field observations and aid the interpretation and design of
field tests.
Chapter 8 summarizes the major work conducted in this dissertation, presents the
main conclusions, and provides some recommendations for future work related to this
research.
16
CHAPTER 2: Understanding Fracture Initiation and Propagation
Pressures1
Fracture-initiation pressure (FIP) and fracture-propagation pressure (FPP) are both
important considerations for preventing and mitigating lost circulation. For significant
fluid loss to occur, a fracture must initiate on an intact wellbore or reopen on a wellbore
with preexisting fractures, and then propagate into the far-field region. Wellbore
strengthening operations are designed to increase one or both of these two pressures in
order to combat lost circulation. Most of the existing theoretical models assume fracture
initiation and propagation pressures are only functions of in-situ stress and rock mechanical
properties. However, as demonstrated by numerous field and laboratory observations, they
are also highly related to drilling fluid properties, and interactions between drilling fluid
and formation rock.
This chapter discusses the mechanisms of lost circulation and wellbore
strengthening, with an emphasis on factors that can affect FIP and FPP. These factors
include: micro-fractures on the wellbore wall, in-situ stress anisotropy, pore pressure,
fracture toughness, filter cake development, fracture bridging/plugging, bridge location,
fluid leak-off, rock permeability, pore size of rock, mud type, mud solid concentration, and
critical capillary pressure. The conclusions of this chapter include information seldom
considered in lost circulation studies, such as the effect of micro-fractures on FIP and the
effect of capillary forces on FPP. Research results described in this chapter may be useful
for lost circulation mitigation and wellbore strengthening design, as well as leak-off test
interpretation.
1 Parts of this chapter have been published in: Feng, Y., Jones, J.F. and Gray, K.E., 2016. A Review on
Fracture-Initiation and-Propagation Pressures for Lost Circulation and Wellbore Strengthening. SPE Drilling
& Completion, 31(02), pp.134-144. This paper was supervised by K. E. Gray.
17
2.1 INTRODUCTION
Most lost circulation events occur when the hydraulic pressure in the wellbore
exceeds FIP and FPP of the formation rock. Lost circulation is common in wellbores with
a narrow drilling mud-weight window, which is the difference between the maximum mud
weight before the occurrence of lost circulation and the minimum mud weight to balance
formation pore pressures or avoid excessive wellbore failure. Typical scenarios include
drilling within depleted reservoirs, drilling highly inclined wellbores where increased fluid
densities are required for hole stability, and drilling highly over-pressured formations,
where the margin between formation pore pressure and the overburden pressure is reduced.
Commonly encountered pressure ramps and pressure regressions may also lead to
significant reductions in the drilling mud-weight window. It is well known that carbonate
formations (limestone/dolomite) are usually characterized by presence of natural fractures,
vugs, and cavities, and consequently lost circulation occurs frequently (Masi et al., 2011;
Wang et al., 2010). However, lost circulation in carbonate formations is outside the scope
of this dissertation and the discussion in this chapter is mainly for clastic formations such
as sandstones and shales.
The reduction in pore pressure in depleted reservoirs results in a corresponding,
albeit smaller, reduction in fracture gradient (Hubbert and Willis, 1957; Matthews and
Kelley 1967). Conversely, bounding and inter-bedded shale layers, as well as any isolated
and un-drained sands, will maintain their original pore pressure and fracture gradient.
Therefore, as shown in the left plot of Figure 2.1, it may be difficult or impossible to reduce
the drilling fluid density sufficiently to maintain equivalent circulating densities (ECD)
below the depleted zone fracture gradient. ECD is defined as the effective density of the
circulating fluid in the wellbore, resulting from the sum of the hydrostatic pressure imposed
by the static fluid column and the friction pressure (API STD 2010). In deep-water
18
formations, the total vertical stress is relatively low since sea water does not provide as
much overburden loading as sediment and rock. A reduction in total vertical stress also
results in a lower lateral stress and fracture gradient. If abnormal pressures are also present,
the mud-weight window may be very narrow, as shown in the right plot of Figure 2.1.
Under these circumstances, it may be challenging to avoid hydraulic fracturing both while
tripping due to surge/swab effects, and while circulating due to high annular friction losses
and ECDs.
Figure 2.1: Left: Pore pressure and fracture gradient plot in depleted zone. Pore pressure
decrease leads to a decrease in fracture gradient. Right: Pore pressure and
fracture gradient plot in deep-water formation with abnormally high pressure.
There is a reduced mud-weight window.
FIP and FPP are two important considerations for preventing and mitigating lost
circulation. Only after a fracture initiates on an intact wellbore or reopens on a wellbore
with preexisting fractures, and then propagates into the far-field region, can significant
fluid loss occur. Therefore, accurate pre-drill estimates of these two pressure values are
critical for reducing lost circulation events. A common theoretical method to estimate FIP
19
for a vertical well, compares a simple tensile failure criterion to the hoop stress defined by
the Kirsch equation. FIP predicted by this approach is related to formation rock strength,
in-situ stresses and the formation fluid pressure, and it is assumed that the fracture initiates
at the wellbore wall. However, the Kirsch equation assumes zero leak-off, i.e. impermeable
rock or perfect mudcake.
FPP can be determined from injectivity tests, analysis of fluid losses while drilling,
or from fracture mechanics modeling. In field practice, FPP is often estimated from leak-
off tests (LOTs), performed at casing or liner shoes. However, these tests are generally
insufficient for this analysis which may lead to significant error (Ziegler and Jones, 2014).
It is worth noting that FIP and FPP are commonly taken as parameters of the
formation rock, dependent on the in-situ stresses, mechanical properties of the rock, and
inclination and orientation of deviated wells. However, field experience suggests they may
also be influenced by other parameters related to the drilling fluid (e.g., mud type, fluid
leak-off, solid particles within the fluid, temperature etc.), as well as other properties of the
rock (e.g., lithology, permeability, wettability and capillary effect). A detailed study of
these factors’ effects on them is therefore needed for better understanding of lost
circulation.
In order to drill through problematic zones with a high risk of lost circulation,
various drilling technologies may be useful, including managed pressure drilling (MPD),
dual gradient drilling (DGD) and casing/liner drilling. Alternatively, “Wellbore
Strengthening” is a different approach that seeks to artificially increase the pressure the
wellbore can sustain and hence widen the mud-weight window. Rather than actually
increase the strength of the wellbore rock, as its name implies, this methodology is believed
to work by plastering wellbore surface and/or bridging/plugging lost circulation fractures.
20
As mentioned in the literature review in Section 1.3, there are two main types of
wellbore strengthening methods currently used in the petroleum industry: the hoop stress
enhancement method, e.g., Stress Cage (Alberty and McLean, 2004), and the fracture
resistance enhancement method, e.g., Fracture-Propagation Resistance (Fuh et al., 1992;
Morita et al., 1990; van Oort et al., 2011). The first method is based on inducing and
plugging a fracture to increase the local hoop stress, thus raising fracture reopening
resistance. Feng et al. (2015a) have conducted detailed numerical studies and found that
theoretically at least, hoop stress can be increased significantly if the fracture can be
plugged effectively. Although theoretical studies show there is large potential in hoop
stress increase (Alberty and McLean, 2004; Wang et al., 2009, 2007b, 2008) and numerous
successes are reported for the Stress Cage method (Aston et al., 2007, 2004a; Song and
Rojas, 2006; Whitfill et al., 2006), lost circulation problems are still commonly
encountered with an ECD much lower than the hoop stress around the wellbore. Therefore
a number of doubts still persist, including (1) whether hoop stress is a good indicator of
lost circulation and the evaluation of wellbore strengthening success and (2) when wellbore
strengthening works, is it actually due to an increase in hoop stress? This chapter will
discuss these questions in detail.
Fracture-Propagation Resistance theory is based jointly on experimental and field
observations, including the DEA 13 (Fuh et al., 1992; Morita et al., 1996a, 1996b, 1990;
Onyia, 1994) and GPRI 2000 (van Oort et al., 2011) laboratory studies. Both theory and
experience indicate fracture-propagation resistance can be effectively enhanced using
appropriate wellbore strengthening methods. Although several models (Fuh et al., 2007;
van Oort et al., 2011; van Oort and Razavi, 2014) have been introduced to explain how
fracture-propagation resistance may be increased, there remains a lack of understanding of
the precise role a list of influencing factors may play. These factors include in-situ stresses,
21
wellbore pressure, fracture geometry and size, mud type and properties, rock lithology and
properties, lost circulation material locations and properties, fluid leak-off, mudcake, and
capillary force. Therefore, significant disagreement about the fundamental physics of
wellbore strengthening still exists in the industry.
The purpose of this chapter is to analyze the mechanisms of lost circulation and
wellbore strengthening, by investigating the factors that may affect both FIP and FPP. In
view of the existing disagreement about the fundamentals of lost circulation and wellbore
strengthening, a critical and detailed analysis of these two pressure thresholds is conducted.
It should be noted that wellbore strengthening discussed in this dissertation is physical or
mechanical strengthening of the wellbore by development of mudcake on wellbore wall
and/or bridging plug in the fracture in relatively permeable formations. In impermeable
shales with very low leak-off, chemical strategies are commonly used to strengthen the
wellbore, either by changing chemical composition of the formation (Growcock et al.,
2009) or by forming chemical sealants in the fracture (Aston et al., 2007). The chemical
wellbore strengthening technique is outside the scope of this dissertation. It should also be
noted that most of the discussions in this chapter are based on the case of a vertical well,
but the principles and perspectives are also applicable to deviated and horizontal drilling.
2.2 LOST CIRCULATION “THRESHOLDS”
For significant fluid loss to occur through either a drilling induced or closed pre-
existing natural fracture, the wellbore pressure must overcome both FIP and FPP. These
two pressure limits may be regarded as “thresholds” to lost circulation, which are critical
for well construction and drilling fluid design.
In theory, FIP is usually greater than FPP, if the wellbore is an intact cylinder.
However, when the stress anisotropy is relatively high and/or there are pre-existing
22
fractures, FPP may be equal to or greater than the calculated FIP. In general, this condition
should not cause significant concern.
There are four general conditions related to lost circulation, depending on the
relative magnitudes of ECD, FIP, and FPP:
(1) When ECD is lower than both FIP and FPP, fluid loss will not occur.
(2) When ECD is higher than FIP but lower than FPP, only very small fractures will
generate near the wellbore wall and no significant fluid loss will occur.
(3) When ECD is larger than FPP but lower than FIP, the situation is less stable. No
fluid loss will occur as long as the wellbore remains intact, and the far field stress
region of each formation is isolated from the pressure in the wellbore. However,
lack of wellbore isolation may result from inadequate filter cake development in
permeable formations or where pre-existing natural or mechanically induced
fractures are present in any type of formation.
(4) When ECD is above both FIP and FPP, fluid loss is expected to occur. In this case,
remedial actions must include some form of ECD reduction and/or wellbore
strengthening operations
2.3 FRACTURE INITIATION PRESSURE (FIP)
Conventional interpretation theories for FIP generally assume a perfectly intact
wellbore. Fracture initiation is predicted when the tangential stress (also called hoop stress)
at the wellbore wall equals the tensile strength of the rock. It is widely accepted that FIP
depends much more on in-situ stresses, which determine the hoop stress around the
wellbore, than the tensile strength of the rock, which is comparatively very small. In reality,
the assumption of a perfectly intact wellbore is rarely true. The most likely imperfect
wellbore condition is a wellbore with micro-fractures (Morita et al., 1990). Micro-fractures
23
may develop naturally from tectonic movement, rapid sediment compaction and/or thermal
fluid expansion, as well as from destructive drilling operations. In the case of pre-existing,
hydraulically conductive micro-fractures at the wellbore wall, the above method to predict
FIP is no longer valid. In this case, the wellbore pressure that begins to fail the formation
rock is the propagation pressure for the micro-fractures, rather than the initiation pressure
for any new fractures. However, for the purposes of this discussion, micro-fracture-
propagation pressure is considered as FIP, since the fracture size is very small and the
assumption of a perfect wellbore is seldom satisfied.
2.3.1 FIP of a Perfect Wellbore
FIP for an intact cylindrical wellbore, may be easily determined from continuum
mechanics (Kirsch equations). However, FIP may be very different for permeable and
impermeable formations. For an impermeable formation with negligible tensile strength,
FIP of a vertical wellbore can be estimated by the Hubbert-Willis equation (Hubbert and
Willis, 1957; Jin et al., 2013):
𝑝𝑖𝑛𝑖 = 3𝑆ℎ𝑚𝑖𝑛 − 𝑆𝐻𝑚𝑎𝑥 − 𝑝𝑝 (2.1)
where, 𝑝𝑖𝑛𝑖 is FIP; 𝑆ℎ𝑚𝑖𝑛 and 𝑆𝐻𝑚𝑎𝑥 are the minimum and maximum horizontal
stresses, respectively; 𝑝𝑝 is the pore pressure.
However, FIP for a permeable rock may be significantly affected by an additional
induced stress term, related to fluid penetration from the wellbore to the formation. For a
permeable rock, FIP can be estimated by the Haimson-Fairhurst equation (Haimson and
Fairhurst, 1967):
𝑝𝑖𝑛𝑖 =3𝑆ℎ𝑚𝑖𝑛−𝑆𝐻𝑚𝑎𝑥−𝜂𝑝𝑝
2−𝜂 (2.2)
𝜂 = 𝛼𝑝 (1−2𝑣
1−𝑣) (2.3)
24
where, 𝜂 is a poroelastic parameter of the rock, which determines the magnitude of the
stress induced by fluid penetration, and various in the range [0, 1], from zero fluid
penetration to unimpeded fluid penetration, respectively; 𝛼𝑝 is Biot’s coefficient; and 𝑣
is Poisson’s ratio.
The left plot of Figure 2.2 shows the relationship between FIP, horizontal stress
anisotropy and pore pressure for a vertical well with a constant poroelastic parameter, 𝜂 =
0.5. It is clear that FIP decreases with an increase in stress anisotropy. It is also clear that
for a given 𝑆ℎ𝑚𝑖𝑛 and 𝑆𝐻𝑚𝑎𝑥 , FIP also decreases with an increase in pore pressure.
However, this observation must be viewed in proper context, since 𝑆ℎ𝑚𝑖𝑛 and 𝑆𝐻𝑚𝑎𝑥 are
generally a function of pore pressure and overburden stress (Hubbert and Willis, 1957;
Matthews and Kelley 1967) and increase with increasing pore pressure, if the overburden
is held constant or increases. With horizontal stress ratio 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛 = 1.3⁄ , the right
plot of Figure 2.2 shows a very interesting observation for the effect of 𝜂 on FIP of a
vertical well. That is, FIP increases with the increase of 𝜂 when the pore pressure is
lower than a certain value but decreases when pore pressure is higher than that value. In
this case, the crossover point is 0.85 ∙ 𝑆ℎ𝑚𝑖𝑛. However, with the decrease of horizontal
stress ratio, the crossover point will move to the right. The crossover point will no longer
exist on the X-axis scale when the horizontal stress ratio is smaller than 1.2.
25
Figure 2.2: Fracture-initiation pressure of a vertical well. Left: with different horizontal
stress anisotropies and pore pressure; Right: with different 𝛈 and pore
pressure.
2.3.2 FIP of a Wellbore with Micro-fractures
As mentioned previously, when hydraulically conductive drilling induced or pre-
existing natural micro-fractures exist on the wellbore, the wellbore pressure that begins to
fail the formation rock is the propagation pressure for the micro-fractures rather than the
initiation pressure for any new fractures. Therefore, the continuum mechanics method
using the Kirsch equation to determine FIP is no longer valid. Instead, a fracture mechanics
approach should be used to determine FIP (or micro-fracture-propagation pressure).
Seeking to interpret leak-off tests for estimating horizontal stress, Lee et al. (2004)
analytically studied the propagation pressure of a fracture extending from a wellbore in the
direction of maximum horizontal stress. This analysis is based on the Barenblatt condition,
which dictates a balance between the tensile stress intensity factor produced by fluid
pressure in the fracture and the negative stress intensity factor caused by the compressive
in-situ stress (Lee et al., 2004; Yew and Weng, 2014). According to their study, the FIP of
a wellbore with micro-fractures (or micro-fracture propagation pressure) should be:
𝑝𝑖𝑛𝑖 =3𝑆ℎ𝑚𝑖𝑛−𝑆𝐻𝑚𝑎𝑥
2+
𝐾𝐼𝐶
𝜋√2𝐿 (2.4)
26
where, 𝐾𝐼𝐶 and 𝐿 are the fracture toughness of the formation and the length of the micro-
fracture, respectively. Based on this equation, FIP is not only related to horizontal stress
but is also a function of fracture toughness 𝐾𝐼𝐶 of the rock, and micro-fracture length 𝐿.
For Eq. 2.4, dimensionally normalizing the pressure and stress terms with minimum
horizontal stress 𝑆ℎ𝑚𝑖𝑛, fracture length with wellbore radius 𝑎, and fracture toughness with
the product of minimum horizontal stress and the square root of wellbore radius 𝑆ℎ𝑚𝑖𝑛√𝑎,
it can be transformed to:
𝑝𝑖𝑛𝑖′ =
3−𝑅
2+
𝐾𝐼𝐶′
𝜋√2𝐿′ (2.5)
where, 𝑝𝑖𝑛𝑖′ , 𝐾𝐼𝐶
′ , 𝐿′ and 𝑅 are dimensionless FIP, dimensionless fracture toughness,
dimensionless fracture length and horizontal stress anisotropy, respectively. Note that Eq.
2.5 has a mathematic singularity signature, as the normalized FIP goes to infinitely high
with a normalized fracture length approaching zero. Dimensional analysis shows that with
reasonable values for R and 𝐾𝐼𝐶′ , Eq. 2.5 is not suitable for a fracture length less than 0.01
inches. In fact, the wellbore can be considered intact, with a fracture as short as 0.01 inches.
The fracture toughness of sedimentary rocks varies approximately in the range of
500 to 2000 psi-in0.5 (Senseny and Pfeifle, 1984; Wang, 2007), and horizontal stress
anisotropy under most geologic settings, ranges from 1 to 2 based on the author’s
experience. With the following assumptions: 𝑆ℎ𝑚𝑖𝑛 = 3000 𝑝𝑠𝑖 , wellbore radius 𝑎 =
4.25 𝑖𝑛, and micro-fracture length 𝐿 = 0.5 𝑖𝑛, Figure 2.3 shows the FIP of a vertical well
under various sets of horizontal stress anisotropy and fracture toughness conditions. It
indicates that FIP (1) is very sensitive to and decreases dramatically with an increase in
horizontal stress anisotropy, (2) increases moderately with an increase in fracture
toughness, and (3) can be much smaller than the minimum horizontal stress with a
relatively high stress anisotropy and low fracture toughness.
27
From this analysis, it is critical to highlight the influence of micro-fractures on FIP.
For instance, in impermeable rocks, the continuum mechanics (Kirsch) equation predicts a
FIP equal to the minimum horizontal stress when stress anisotropy is 2.0. However, with
the same stress anisotropy, Figure 5.3 shows FIP will be far below the minimum horizontal
stress, with a fracture length of only about 10% of the wellbore radius.
Figure 2.3: FIP (micro-fracture-propagation pressure) decreases dramatically with an
increase in horizontal stress anisotropy, and increases moderately with an
increase in fracture toughness; it can be much smaller than the minimum
horizontal stress with high stress anisotropy and low fracture toughness.
2.3.3 FIP versus Leak-off Pressure (LOP)
In conventional field practice, leak-off tests are often used to estimate FIP, which
is taken as the pressure value at the first inflection point where the pressure ramp-up curve
deviates from linearity before formation breakdown. A typical pressure-volume/time
response of a leak-off test is shown in the middle figure of Figure 2.4.
Although it is commonly accepted that the leak-off pressure (LOP) indicates the
start of a fracture and should be identical to the FIP, a careful analysis indicates they are
28
not necessarily the same, especially when “dirty” mud (drilling fluid with high solids
content) is used for a leak-off test in a permeable formation.
For an intact wellbore with solids-free fluid or clean mud, fracture initiation is
largely dominated by in-situ stresses. For a wellbore with micro-fractures and clean mud,
fracture initiation is controlled by the fluid pressure distribution inside the fracture. The
LOP in these two cases should be approximately equal to FIP. However, for a drilling fluid
with high solids content, e.g. LCMs, the mud properties may affect the observed leak-off
behavior and lead to a LOP very different from FIP. This can be explained as follows.
When a short hydraulically conductive micro-fracture is created during a leak-off
test, in theory it should be easily extended with sufficient wellbore pressure. In reality, the
micro-fracture may be quickly sealed by mud solids, forming a filter cake within the
fracture. This “internal” filter-cake can then isolate the fracture from the wellbore, and not
enough fluid pressure will reach the fracture face in order to extend it. This “opening and
healing” or “fracturing and packing” behavior within the fracture, can theoretically restore
the pressure containment capability of the wellbore, and perhaps increase it to a higher
value than the ideal case where no fracture exists.
This phenomenon is similar to wellbore strengthening. However, the “opening and
healing” of such small fractures is not likely detectable in a field leak-off test or even in a
lab test (Guo et al., 2014). In many field leak-off tests, it is difficult to identify a clear leak-
off response at fracture-initiation point, and the FIP can be very close to the formation
breakdown pressure (FBP) as shown in the left figure of Figure 2.4. Therefore, the lack of
a visible leak-off response reasonably below the FBP doesn’t necessarily mean a small
fracture has not been generated.
Numerous elements may influence the signature of a leak-off test, including the
compressibility and elasticity of the mud, casing, cement and formation rock, fluid seepage
29
from the wellbore wall, and fluid leak-off into fractures. Among these factors, only the
effect of leak-off into fractures is observably nonlinear (Fu, 2014). Therefore, when there
is a clear leak-off response as shown in the middle figure of Figure 2.4, a relatively large
fracture is likely to have been created, and the leak-off point, commonly considered to be
fracture initiation, is actually micro-fracture propagation. Undetectable micro-fracture
generation has already occurred before this leak-off point, so LOP is somewhat higher than
FIP.
It is also possible to observe multiple leak-off points on the pressure-volume/time
curve. The right figure of Figure 2.4 shows a case where there are two inflection points.
This signature is more common for leak-off tests conducted in permeable formations, with
a low pump rate and high solids content fluid. These observations may be explained by a
filter cake break within the fracture, where wellbore pressure breaks the filter cake, leading
to additional fracture extension. The fracture will be quickly sealed again by solids in the
mud, and the wellbore pressure will continue to build. If the subsequent wellbore pressure
increases enough, the filter cake may fail again, and the process is repeated. This repeated
fracturing and healing behavior might continue until formation breakdown. It should be
emphasized that a clear slope-change in the pressure-volume/time response during a leak-
off test is usually subsequent to fracture initiation. This response is most likely a filter cake
break in a fracture larger than a micro-fracture, but still in the vicinity and under the
influence of the near wellbore stress concentration. Lab tests show that fractures can grow
significantly without any clear leak-off signature (Guo et al., 2014).
It may not be possible to accurately predict FIP from a leak-off test using a high
solids content fluid. A slope-change point may be undetectable prior to formation
breakdown, or if detected, it may indicate filter cake breakdown rather than fracture
initiation.
30
Figure 2.4: Schematic pressure-volume/time curves in leak-off tests. Left: no visible leak-
off response at fracture initiation, the leak-off pressure is very close to
formation breakdown pressure; Middle: a clear leak-off point before
formation breakdown; Right: multiple leak-off points before formation
breakdown.
2.4 FRACTURE PROPAGATION PRESSURE (FPP)
After initiation, a fracture will tend to propagate from the wellbore wall to the far-
field, under sufficient wellbore fluid pressure. Typically, this fracture propagation consists
of both a stable and an unstable stage. During a leak-off test, the stable fracture-propagation
stage begins at fracture initiation or leak-off and ends roughly at formation breakdown.
Initially the fracture grows very slowly and its volume increases at a rate lower than the
pump rate. Therefore, the wellbore pressure continues to rise prior to formation breakdown,
which is the upper pressure limit for stable fracture growth.
The unstable fracture-propagation stage begins immediately following formation
breakdown. Over a very short time period, the fracture volume expands at a much greater
rate than the pump rate and the wellbore experiences a sudden pressure drop. Ultimately,
the wellbore pressure stabilizes as fracture propagation continues, with a rate of fracture
volume increase roughly equal to the pump rate. From a theoretical viewpoint, the FPP
with clean injection fluid will gradually decrease with the continued increase in fracture
31
length, as will be shown later in the chapter. However, from a practical viewpoint, the FPP
can either increase or decrease with fracture growth, likely due to the high friction pressure
in a relatively large fracture and the complex nature of formation rock.
FPP is a very important parameter for well construction and drilling fluid design,
especially for lost circulation prevention. In challenging areas with severe lost circulation
problems, extended leak-off tests (XLOTs) are recommended in order to obtain reliable
estimates of FPP.
2.4.1 Formation Breakdown Pressure (FBP)
With a clean fluid, the pressure required to initiate a fracture on the wellbore wall
is usually greater than that required to propagate the fracture into the formation.
Furthermore, formation breakdown is often assumed to occur when the hoop stress at the
wellbore wall equals the tensile strength of the rock (Hubbert and Willis, 1957).
Using a fluid with a high solids content, numerous laboratory and field tests
(Aadnøy and Belayneh, 2004; Guo et al., 2014; Liberman, 2012; Morita et al., 1990) have
shown that FBP is often significantly higher than that predicted by conventional continuum
mechanics theories. This phenomenon may be elegantly explained by the filter cake sealing
effect.
Prior to formation breakdown, the fracture size (length) remains small, and fracture
propagation is determined by fracture toughness. When the fracture length is small, the
toughness term in Eq. 2.5 can be much larger than the stress term. According to linear
elastic fracture mechanics, a tensile fracture will start to extend when the stress intensity
factor 𝐾𝐼 reaches fracture toughness 𝐾𝐼𝐶, i.e.
𝐾𝐼 = 𝐾𝐼𝐶 (2.6)
32
The stress intensity factor 𝐾𝐼 is a function of fracture size and geometry, as well as
load condition. Fracture toughness 𝐾𝐼𝐶 is a material constant representing the strength of
the material. For a short fracture on the wellbore wall as shown in Figure 2.5, based on
linear elastic fracture mechanics theory, the stress intensity factor can be estimated by:
𝐾𝐼 = 1.12(𝑃𝑓 − 𝑆𝜃𝜃̅̅ ̅̅̅)√𝜋𝐿 (2.7)
where, 𝑃𝑓 is the pressure inside the fracture; 𝑆𝜃𝜃̅̅ ̅̅̅ is the average normal stress (closure
stress) acting on the fracture face and can be roughly calculated by the Kirsch equation
(neglecting the presence of the fracture). Hence, for a given fracture, 𝑆𝜃𝜃̅̅ ̅̅̅ is only a function
of the wellbore pressure and the far-field stresses. In most geologic settings, 𝑆𝜃𝜃̅̅ ̅̅̅ is a
compressive stress, unless a very high horizontal stress anisotropy (larger than 3) exists. In
order for 𝐾𝐼 to reach 𝐾𝐼𝐶 to propagate the fracture, 𝑃𝑓 must be large enough to overcome
the closure stress 𝑆𝜃𝜃̅̅ ̅̅̅. Therefore, for a given fracture, wellbore pressure and horizontal
stresses, 𝑃𝑓 acting on the fracture face should dominantly control fracture propagation.
When the fracturing fluid is clean, wellbore fluid can easily flow into the fracture,
and apply pressure to the fracture face, approximately the same magnitude as wellbore
pressure. Thus, a stress intensity factor higher than the fracture toughness is more easily
achieved and the fracture will propagate. However, when the fluid contains solids, the
following mechanisms will significantly reduce or eliminate the pressure acting on the
fracture face, preventing fracture propagation:
Solids are transported with fluid flow into the fracture, resulting in a high solids
density and fluid viscosity in the fracture. The fracture may also be plugged/sealed
by a filter cake, as shown in the middle figure of Figure 2.5. A high solids density
and/or fluid viscosity will significantly increase the fracture pressure drop from
fracture inlet to tip, leading to a much lower 𝑃𝑓 and smaller 𝐾𝐼. Filter cake sealing
inside the fracture can further decrease 𝑃𝑓 as well as 𝐾𝐼.
33
Due to the small aperture of the fracture, it is very likely to be bridged and sealed
quickly by the filter cake, before solids can enter the fracture, as shown in the right
figure of Figure 2.5. The low permeability of the filter cake will further restrict fluid
flow into the fracture, and finally lead to 𝑃𝑓 in the fracture equal to pore pressure,
due to pressure bleed-off into porous rock. The excess pressure (𝑃𝑓 − 𝑆𝜃𝜃̅̅ ̅̅̅) will
then decrease or become negative under most conditions. Therefore, the stress
intensity factor will not reach the fracture toughness magnitude, unless the wellbore
pressure builds high enough to break the filter cake at the fracture mouth. As
mentioned previously, the “fracturing and healing” process can be repeated several
times prior to formation breakdown, and therefore the FBP may be significantly
higher than the theoretically predicted FIP.
Figure 2.5: A small fracture on the wellbore wall before formation breakdown: Left: no
filter cake plugging with clean fluid; Middle: high solids concentration or
filter cake inside the fracture; Right: fracture is plugged at the inlet on
wellbore wall.
34
2.4.2 Fracture Propagation Pressure (FPP)
2.4.2.1 Theoretical Prediction
At formation breakdown, the filter cake in the micro-fracture breaks completely,
allowing fluid to enter the fracture. The fracture then grows quickly in both width and
length, extending to the far-field region. The wellbore pressure drops to the FPP. A great
number of field and lab hydraulic fracturing tests have indicated that FPP decreases with
the increase in fracture length. This phenomenon might be partly due to the minimal excess
pressure required to maintain fracture propagation with a large fracture face, and partly due
to the high accessibility of a weak point, with a large fracture circumference (Okland et al.,
2002). However, this phenomenon can be interpreted more elegantly with a coupled fluid
mechanics and solid (fracture) mechanics approach.
After the fracture has propagated a significant distance, the influence of the
wellbore on fracture propagation behavior is greatly diminished (Zheltov, 1955). Figure
2.6 shows a hydraulic fracture with a neglected wellbore in the fracture center. Consider a
fracture with a length of 2𝐿, perpendicular to the minimum horizontal stress 𝑆ℎ𝑚𝑖𝑛, as
shown in Figure 2.6. The formation rock is considered isotropic, homogeneous, linearly
elastic, and impermeable. The fluid is assumed to be incompressible, non-viscous
Newtonian fluid. It is injected through the well at fracture center at a constant rate 𝑄. The
pressure everywhere inside the fracture is the same as wellbore pressure. Using a coupled
fluid and solid mechanics method, similar to that of Detournay (2004), both the fracture
half-length and pressure during fracture propagation can be determined as functions of
time:
𝑎(𝑡) = (𝐸′𝑄(𝑡−𝑡0)
2𝜋1 2⁄ 𝐾𝐼𝐶)
2 3⁄
∝ 𝑡2 3⁄ (2.8)
𝑝(𝑡) =𝐸′𝑄(𝑡−𝑡0)
2𝜋𝑎2∝ 𝑡−1 3⁄ (2.9)
35
where 𝑡 is injection time; 𝑎(𝑡) is the fracture half-length at time 𝑡; 𝑝(𝑡)is the pressure
inside the fracture at time 𝑡; 𝐸′ =𝐸
1−𝑣2 is the plane strain modulus, which is a function of
Young’s modulus 𝐸 and Poisson’s ratio 𝑣 ; and 𝑡0 is the start time of fracture
propagation.
The pressure behavior theoretically predicted by Eq. 2.9 is schematically shown in
the left plot of Figure 2.7. Before 𝑡0, the pressure builds up linearly inside the fracture
without fracture propagation. At 𝑡0 , the stress intensity factor of the fracture reaches
fracture toughness, triggering sudden fracture propagation. Following 𝑡0 , the pressure
drops nonlinearly and proportional to 𝑡−1/3. Figure 2.7 is the FPP when water based mud
was used as the fracturing fluid in a lab test of DEA-13 project (Fuh et al., 1992; Morita et
al., 1990). Apart from the fluctuating signature, a fitted curve shows the pressure decreases
proportionally to 𝑡−0.305, which is reasonably close to the predicted result.
Figure 2.6: A large hydraulic fracture (wellbore at the fracture center is neglected).
36
Figure 2.7: Pressure response during fracture propagation. Left: theoretical result; Right:
DEA-13 lab test result.
The above model is established under the assumptions of an ideal condition: the
rock is impermeable and the fluid is clean with zero viscosity. Eq. 2.9 shows FPP for the
model depends only on injection rate, injection time, and fracture toughness. In reality,
FPP also depends on a list of other factors including in-situ stress, pore pressure, solids
plugging, base fluid leak-off, lithology, permeability, aqueous/non-aqueous fluid, rock
wettability, capillary force and others. Most of these factors’ effects are not independent,
but related to others. Several of these factors’ effects on FPP are discussed as follows.
2.4.2.2 In-situ Stress, Pore Pressure and Solids Plugging
Consider a fracture similar to that in Figure 2.6, which is perpendicular to the
minimum horizontal stress, but now the fracture is in a formation with pore pressure 𝑝𝑝,
and is effectively plugged by solid particles in the fracturing fluid at some location inside
the fracture as shown in Figure 2.8. Assume the plug is perfect without permeability, thus
it completely stops fluid penetration. The fracture domain ahead of the plug, from the
wellbore to the plug, is wetted by fluid and its pressure is the same as wellbore pressure.
The pressure in the fracture section behind the plug (non-penetrated zone) is equal to pore
37
pressure, due to fracture pressure bleed-off into porous rock. This problem was first solved
analytically by Abé et al. (1976). On the basis of their work, the FPP for a large fracture
can be given roughly by the following equation:
𝑝𝑝𝑟𝑜𝑝 =1
1−√1−(1−𝐿𝑛𝑤
𝐿)
2[𝑆ℎ𝑚𝑖𝑛 − 𝑝𝑝√1 − (1 −
𝐿𝑛𝑤
𝐿)
2
] (2.10)
where, 𝑝𝑝𝑟𝑜𝑝 is the FPP; 𝐿𝑛𝑤 is the fracture length of the non-penetrated zone. Note that
Eq. 2.10 is only valid for a fracture with a length much larger than the wellbore radius.
Therefore, the effect of the wellbore can be ignored. Another limitation of this equation is
that it should not be used when the plug location is close to the wellbore, because the
detailed stress-concentration in the wellbore vicinity is neglected. Eq. 2.10 also neglects
the effect of fracture toughness due to its unimportant role when the fracture is large (this
is one of the major differences between large fracture and micro-fracture propagation: the
influence of fracture toughness might be the dominate factor for micro-fractures, but trivial
for large fractures).
It is indicated by Eq. 2.10 (also see Figure 2.9) that FPP after plugging is primarily
determined by the minimum horizontal stress, pore pressure and the non-penetrated zone
length or the location of the plug. As shown in Figure 2.9, for a given minimum horizontal
stress 𝑆ℎ𝑚𝑖𝑛 , with an increase in pore pressure 𝑝𝑝 , FPP decreases. Another important
observation from Figure 2.9 is that for low values of 𝑝𝑝 𝑆ℎ𝑚𝑖𝑛⁄ , corresponding to
formations with hydrostatic or abnormally low pressure, the FPP is very sensitive to non-
penetrated zone size, the larger the non-penetrated zone size, the higher the FPP. This
confirms the statement in the Stress Cage concept (Alberty and McLean, 2004; Feng et al.,
2015a) that the best place to plug a fracture for wellbore strengthening is the fracture inlet
or mouth, and also the statement in the Fracture-Propagation Resistance concept (van Oort
38
et al., 2011) that plugging the fracture to isolate its tip from wellbore pressure can
significantly enhance the fracture propagation resistance (pressure). FPP in low pressure
formations, as shown in Figure 2.9, can be increased to several times higher than the
minimum horizontal stress. However, the influence of plugging is smaller for high values
of 𝑝𝑝 𝑆ℎ𝑚𝑖𝑛⁄ , e.g. formations with abnormally high pressure. Therefore, from the pore
pressure point of view only, wellbore strengthening methods based on plugging the fracture
might be more effective for depleted reservoirs with larger differences between 𝑝𝑝 and
𝑆ℎ𝑚𝑖𝑛than for high-pressure formations with relatively small differences between 𝑝𝑝 and
𝑆ℎ𝑚𝑖𝑛.
Figure 2.8: A fracture plugged by solid particles.
39
Figure 2.9: FIP with non-penetration zone length, pore pressure and minimum horizontal
stress.
2.4.2.3 Fluid Leak-off through Fracture Faces
In order to investigate the effect of fluid leak-off on fracture propagation behavior,
a stationary fracture model as shown in Figure 2.10 is used. In a hypothetical fracture
extending perpendicular to the minimum horizontal stress direction, in a poroelastic rock
with initial pore pressure 𝑝𝑝 and time 𝑡 = 0, a fluid pressure 𝑃𝑓 (greater than 𝑝𝑝) is
applied inside the fracture. The fracture fluid and pore fluid have identical properties.
Therefore, after applying fluid pressure, the normal traction on the fracture face changes
from 𝑆ℎ𝑚𝑖𝑛 to −𝑃𝑓 (here tension is positive) while the pore pressure on the fracture face
changes from 𝑝𝑝 to 𝑃𝑓. Detournay and Cheng (1991) indicated that this problem can be
examined by decomposing it into two separate problems (1) applying normal traction (fluid
pressure) to the fracture face while keeping pore pressure unchanged, and (2) applying pore
pressure while keeping the traction constant. The solutions of each problem are then
40
superposed to obtain the full solution of the original problem. In this case, only problem 2
is of interest (the effect of pore pressure increase on fracture behavior, due to fluid leak-off
thorough the fracture face). According to the analysis results reported by Detournay and
Cheng (1991), a pore pressure increase in this case will lead to a negative change in both
fracture volume and stress intensity factor. As schematically shown in Figure 2.11, the
instantaneous fracture volume 𝑉𝐶0 at 𝑡 = 0 decreases to the long-term volume 𝑉𝐶∞ ,
when pore pressure reaches 𝑃𝑓. The instantaneous stress intensity factor 𝐾𝐼0 also drops to
the long-term value 𝐾𝐼∞ . The decrease of fracture volume and stress intensity factor
reveals the fact that a pore pressure increase as a result of fluid leak-off tends to close the
fracture and inhibit fracture growth.
Figure 2.10: A stationary fracture model.
41
Figure 2.11: Fracture volume (left) and stress intensity factor (right) changes with
applying pore pressure to the fracture face only, while keeping the traction
on the fracture face constant.
2.4.2.4 Permeability
The above analyses of fracture propagation indicate that both solids plugging and
fluid leak-off induced pore pressure increases can contribute to preventing fracture
propagation or enhancing fracture-propagation resistance. Fluid leak-off, however, is well
recognized as a critical prerequisite for creating an effective filter plug (Aston et al., 2007).
Therefore, any factors affecting filter plug formation and/or fluid leak-off can influence
fracture propagation, hence lost circulation and wellbore strengthening.
Permeability attracts much attention in lost circulation and wellbore strengthening
analysis, since it is generally believed that only in permeable formations (i.e. sandstone)
can an effective filter plug be formed. Conversely, in impermeable rocks (i.e. shale) it is
generally believed that wellbore strengthening is not likely to be successful, although
several successful cases in shale are reported using specific pre-engineered drilling fluids
and LCMs (Aston et al., 2007). When permeability is low as depicted in the left figure of
Figure 2.12, the base fluid (or filtrate) leak-off rate is too low to allow mud solids or LCMs
to aggregate in the fracture and therefore an effective filter plug is not formed. Low leak-
42
off rates also mean very limited fracture pressure/energy is released into the formation.
Therefore, pressure is trapped inside the fracture, facilitating fracture growth. On the
contrary, in permeable formations as depicted in the right figure of Figure 2.12, filtrate
leak-off rate is high enough to form an effective filter plug, and the pore pressure increase
due to fluid leak-off inhibits fracture growth as previously discussed. In addition, fracture
pressure/energy is easily released into the formation, hence less pressure/energy acts
toward extending the fracture.
Figure 2.12: Hydraulic fractures in impermeable (left) and permeable (right) formations
with high solids content fluid.
2.4.2.5 Capillary Entry Pressure
Additionally, if the wellbore fluid (filtrate) and pore fluid are immiscible, capillary
entry pressure 𝑃𝑐𝑒 , also known as threshold capillary pressure, is an important
consideration for analyzing fluid leak-off behavior, especially if pore throat openings or
capillaries are relatively small (Nelson, 2009). Unfortunately, this parameter is often
neglected for lost circulation mitigation and wellbore strengthening design. High 𝑃𝑐𝑒 can
significantly inhibit fluid leak-off, filter-cake/plug development and pore pressure
increase. 𝑃𝑐𝑒, usually estimated by the Young-Laplace equation, depends highly on the
43
largest pore opening (throat) size, wettability (contact angle) and miscibility (interfacial
tension) of the drilling and pore fluids.
𝑃𝑐𝑒 = 2𝛾𝑓,𝑚1
𝑟𝑐𝑜𝑠𝜃 (2.11)
where, 𝑃𝑐𝑒 is the capillary entry pressure, 𝛾𝑓,𝑚 is the interfacial tension between the
wellbore fluid and pore fluid, 𝑟 is the largest pore opening radius, and 𝜃 is the wetting
(contact) angle.
It is clear from Eq. 2.11 that 𝑃𝑐𝑒 increases as pore opening size decreases. When
the difference between wellbore pressure and pore pressure exceeds 𝑃𝑐𝑒, wellbore fluid
(filtrate) will be pushed (leak-off) into the formation and displace the pore fluid. The typical
pore opening size for sandstone is from several to dozens of microns, but much smaller for
shale, in the range of several to dozens of nanometers (Nelson, 2009). In sandstone, 𝑃𝑐𝑒
for hydrocarbons and brine is roughly 10 to 50 psi, and for shale it is roughly 200 to 800
psi in deepwater Gulf of Mexico according to the study by Dawson (2004) and Dawson
and Almon (2005). Since sandstone has significantly lower 𝑃𝑐𝑒 than shale, it is easier for
the wellbore fluid to leak-off into sandstone, facilitating filter plug development and pore
pressure elevation. In contrast, this is less likely to happen in shale due to its high 𝑃𝑐𝑒.
Figure 2.13 schematically shows the fluid leak-off and the corresponding filter-cake/plug
development controlled by capillary pressure for water-wet sandstone and shale when the
fracture and pore fluids are oil/synthetic based mud (OBM/SBM) and water (brine),
respectively.
44
Figure 2.13: Fluid leak-off and filter plug development controlled by capillary pressure
for water-wet sandstone with larger pore size and shale with smaller pore
size. Formation fluid is water while fracture fluid is oil/synthetic based mud.
In addition to pore opening size, rock wettability and fluid immiscibility also
control 𝑃𝑐𝑒, and therefore fluid leak-off and filter plug formation. It is also important to
note that for extremely low permeability shale, fluid leak-off may be very restricted
regardless of fluid type. For brine saturated, water-wet rocks with relatively small pore
opening size: (1) If the fluid is OBM/SBM, it cannot easily enter the pore openings due to
high interfacial tension between immiscible fluids, and therefore there is little if any fluid
leak-off or filter-cake/plug development (Figure 2.14); (2) in contrast, if the fluid is WBM,
the water in the mud may readily invade the pore openings, leaving the solid particles
behind and thereby forming a filter-cake/plug.
45
Figure 2.14: Fluid leak-off and filter cake development controlled by fluid immiscibility
and capillary pressure.
The above capillary entry pressure analysis may further explain the following field
observations:
(1) Lost circulation in fractured and silty shale formations occurs much more
frequently with OBM/SBM than with WBM, and it is often more difficult to cure
fluid losses with OBM/SBM. Due to high capillary entry pressures, OBM/SBM
cannot easily invade the pores of water-wet shale and silty shale (most shale is
water-wet), and therefore, all the fluid pressure acts toward propagating the fracture
tip. No effective filter plug is developed to isolate wellbore pressure and increase
fracture propagation resistance.
(2) Wellbore “breathing” is a phenomenon that occurs when formations take drilling
fluid when the pumps are on and give the fluid back when the pumps are off, due
to the opening and closing of drilling induced fractures. This phenomenon is usually
observed in water-wet shale (especially silty-shale) formations, while drilling with
OBM/SBM. One plausible explanation is that OBM/SBM will often flow back to
the wellbore rather than leak-off into the formation, due to very high capillary entry
46
pressures (Ziegler and Jones, 2014). Conversely, WBM will leak-off readily into
these same formations, rather than flow back to the wellbore.
2.5 PREVENTIVE AND REMEDIAL WELLBORE STRENGTHENING
There are two kinds of wellbore strengthening treatments in the drilling industry:
preventive wellbore strengthening and remedial wellbore strengthening. Simply put,
preventive wellbore strengthening methods attempt to “strengthen” the wellbore to prevent
fluid loss due to hydraulic fracturing. Remedial wellbore strengthening methods attempt to
“strengthen” the wellbore after fluid loss due to hydraulic fracturing has already occurred.
Preventive methods focus on increasing both FIP and FPP, while remedial methods focus
primarily on FPP, since a fracture has already been created.
In a preventive wellbore strengthening treatment, LCM has a dual purpose for
preventing fracture initiation. First, LCM helps develop a mud filter cake with low
permeability and high ductility. As discussed previously, this filter cake will help maintain
a high FIP, by effectively isolating the formation from pressure in the wellbore, thus
inhibiting any pore pressure increase in the vicinity of the wellbore wall.
Second, LCM particles can immediately plug any generated micro-fractures,
preventing both fluid flow into the fracture and pressure communication between the
wellbore and fracture tip. In theory, this process should restore (maintain) a leak-off
pressure higher than FIP. This claim can be confirmed by the experimental wellbore
strengthening study by Guo et al. (2014).
Figure 2.15 shows the pressure build-up curve, when using a drilling fluid with 20
ppb graphitic LCMs, during a preventive treatment test (Guo et al., 2014). The pressure
was increased to 2500 psi (test device limit) without apparent leak-off. Since no leak-off
response was observed, it might be concluded that the formation had not been fractured. In
47
fact, the test block was fractured completely to its edges (see Figure 2.16). A reasonable
explanation for this observation is that fractures initiated on the wellbore wall were
immediately sealed by LCMs in the mud. Therefore, no significant mud loss or noticeable
pressure response occurred. This test demonstrates that observed leak-off pressure is not
necessarily equal to FIP, especially when LCM is used. Leak-off pressure can be somewhat
higher than FIP and fracture initiation can occur without fluid loss. This test also confirms
that leak-off pressure can be significantly increased by using LCMs in a preventive
wellbore strengthening treatment.
After fracture initiation and fluid loss have occurred, preventive wellbore
strengthening treatments work in the same manner as remedial treatments. With fluid loss,
LCM particles are forced into the fracture to form a solid bridge or filter cake within the
fracture, and thereby increase fracture propagation resistance. However, fracture initiation
and leak-off pressures are not generally restored to their original values by remedial
treatments.
This assertion is confirmed by the DEA-13 experimental study (Black et al., 1988;
Onyia, 1994) (see Figure 2.17). This repeated test included 3 cycles. In the first cycle the
intact wellbore was fractured with a fluid containing LCMs, and a relatively high leak-off
pressure was observed. The second cycle was a repeat of the first cycle, but without LCMs.
In this case, leak-off pressure was much lower than in the first cycle, because the mud
barrier on the wellbore wall was previously destroyed. FPP was also much lower since
there was no effective bridge or filter cake development without LCM. In the third cycle,
LCM was added back to the fluid, simulating a remedial wellbore strengthening treatment.
In this case, leak-off pressure did not change compared to the second cycle. However, FPP
did significantly increase.
48
From this analysis, it can be claimed that both FIP and FPP can be increased by
wellbore strengthening treatments. Preventive treatments can increase both pressures,
while remedial treatments only alter FPP. However, significant mud loss in a drilling well
will only occur if FPP is exceeded. It is also worth repeating, that these treatments work
much better in more permeable formations with low capillary entry pressures. An
illustration of both methods is shown in Figure 2.18.
Figure 2.15: An example of a preventive wellbore strengthening test on a sandstone block
(after Guo et al., 2014).
49
Figure 2.16: The sandstone test block for pressure build-up curve in Figure 12. The block
was fractured to the edges without obvious fluid leak-off due to LCM
sealing effect (after Guo et al., 2014).
Figure 2.17: A repeated hydraulic fracturing test with LCM. First injection cycle
(preventive treatment - intact wellbore, with LCM): high leak-off pressure
and high propagation pressure. Second injection cycle (fractured wellbore,
without LCM): low leak-off pressure and low propagation pressure. Third
injection cycle (fractured wellbore, with LCM): low leak-off pressure, high
(increased) propagation pressure. (after Black et al., 1988).
50
Figure 2.18: Preventive wellbore strengthening treatment enhances both leak-off pressure
and FPP, while remedial wellbore strengthening treatment only enhances
FPP.
2.6 SUMMARY
FIP of a perfectly cylindrical wellbore can be determined by continuum mechanics
methods (Kirsch equations). However, for a wellbore with micro-fractures, fracture
mechanics methods should be used to predict FIP.
FIP of a wellbore with micro-fractures is controlled not only by pore pressure and
in-situ stresses, but also by fracture length and fracture toughness of the formation
rock. It can be much lower than that of a perfect wellbore.
Leak-off pressure from a leak-off test may not be equivalent to the FIP when a high
solids content “dirty” mud is used. Due to the continuous sealing effect of dirty
mud, the observable “leak-off” pressure may instead be the filter cake breakdown
pressure (i.e. propagation pressure) of a relatively larger sealed fracture, rather than
FIP of an intact wellbore wall.
51
Formation-breakdown pressure is the upper pressure limit for the stable fracture
propagation stage. During this stage, the fracture size remains small, and a fracture
mechanics method can be used to determine formation-breakdown pressure, which
is controlled to a very large extent by fracture toughness. A high solids
concentration in the drilling fluid, a filter-cake inside the fracture and/or a filter
plug at the fracture mouth can significantly increase formation-breakdown
pressure.
FPP is the fracture pressure during the unstable propagation stage. A coupled fluids
and solids mechanics method predicts a decrease in FPP with an increase in fracture
length.
Plugging a fracture can significantly increase its propagation pressure, especially
in formations with large differences between pore pressure 𝑃𝑝 and minimum
horizontal stress 𝑆ℎ𝑚𝑖𝑛 . Therefore, wellbore strengthening methods based on
plugging the fracture should be more effective in depleted reservoirs with large
differences between 𝑃𝑝 and 𝑆ℎ𝑚𝑖𝑛 than in deepwater over-pressured formations
with relatively small differences between 𝑃𝑝 and 𝑆ℎ𝑚𝑖𝑛.
Fluid leak-off through the fracture face hinders fracture growth by facilitating filter
cake development and reducing the fluid energy available to propagate the fracture.
Capillary entry pressure 𝑃𝑐𝑒 is an important and often neglected consideration for
lost circulation mitigation and wellbore strengthening. High capillary entry
pressures, associated with small pore openings and immiscible fluids, can
significantly restrict fluid leak-off and filter-cake/plug development. Field
observations indicate lost circulation in fractured and silty shale formations occurs
more frequently with OBM/SBM than with WBM. Additionally, the observation
52
that wellbore breathing typically occurs in water-wet formations drilled with
OBM/SBM may be elegantly explained by capillary theory.
A preventive wellbore strengthening treatment can increase both FIP and FPP,
while a remedial treatment can only increase FPP.
53
CHAPTER 3: Developing a Framework for Lost Circulation Simulation
Understanding the growth of drilling induced fracture is critical for lost circulation
prevention and remediation. It can help provide useful information for optimization of
drilling fluid rheology, well configuration, pump schedule, and LCM particle size
distribution. In this chapter, a lost circulation simulation framework is developed based on
finite-element method using Abaqus®. The framework can be used to simulate static fluid
loss without mud circulation in the wellbore and dynamic fluid loss with mud circulation.
The framework consists of two components: wellbore and formation rock. It
successfully couples the fluid circulation in the wellbore, fluid seepage on wellbore wall,
propagation of induced fractures, fluid flow in induced fractures, pore fluid flow and
deformation of formation rock. The fluid circulation in the wellbore is modeled based on
the Bernoulli’s equation taking into account gravitational and viscous pressure losses of
fluid flow. Fracture propagation and fluid flow in the fracture are modeled based on a pore
pressure cohesive zone method. A traction-separation constitutive law for describing
fracture propagation and a fluid flow constitutive law for describing fluid flow in the
fracture are incorporated into the cohesive zone model. Fluid seepage on wellbore wall,
pore fluid flow and porous rock deformation are modeled using poroelastic theory. The
numerical model provides a novel way to simulate fluid losses during drilling when the
boundary condition at the fracture mouth is neither a constant flowrate nor a constant
pressure, but rather a dynamic wellbore pressure.
54
3.1 INTRODUCTION
Understanding drilling induced fracture behavior can provide an effective tool for
guiding equivalent circulation density (ECD) optimization and LCM selection for lost
circulation prevention. Traditionally, analytical models are used to predict the wellbore
pressure or ECD at which fractures begin to initiate on wellbore wall (Kostov et al., 2015).
As mentioned in Section 2.3, there are two commonly used analytical models. The first one
is the Hubbert-Willis model (Hubbert and Willis, 1957) which does not consider the effect
of fluid diffusion from wellbore to surrounding formation and usually predicts an upper
limit for fracture initiation pressure (FIP). The other one is the Haimson-Fairhurst model
(Haimson and Fairhurst, 1967) that considers the effect of fluid diffusion and usually
predicts a lower limit for FIP. While these analytical models can provide reasonable
prediction for FIP, they cannot give any information about the fracture pressure and
fracture dimensions once a fracture has been induced while drilling.
Different from injectivity tests or hydraulic fracturing stimulation treatments in
which fluid driven fractures are intentionally created, most of the drilling induced fractures
are unintentional. Even though there are a lot of numerical simulation studies on hydraulic
fracturing in the literature, very few studies are performed on modeling drilling induced
fractures. While a fracture model with pre-designed constant or time-dependent injection
rate can capture the fracture behaviors in an injectivity test or hydraulic fracturing
operation, it cannot describe the fracture induced during drilling. Drilling induced fractures
are “dynamic-pressure-driven”, rather than “constant-rate-driven” or “constant-pressure-
driven”. In other words, there is neither a constant flow rate nor a constant pressure at the
fracture mouth, but rather a dynamic pressure or ECD in the bottom hole which drives
fracture propagation.
55
The dynamic bottom hole pressure (BHP) or ECD while drilling is influenced by
the gravity pressure of the fluid column and viscous pressure loss of fluid flow in the
annulus. For a well with a certain depth, gravity pressure is dominated by drilling mud
density, while viscous loss is controlled by a number of factors including mud density, mud
viscosity, flow (pump) rate, annulus clearance, and roughness of annulus surfaces. These
factors, together with fluid flow and fracture propagation in the formation, are successfully
coupled in the lost circulation simulation framework developed in this chapter.
The lost circulation model development is described in the below sections. Section
3.2 provides the numerical method and governing equations for wellbore fluid flow,
fracture propagation, fracture fluid flow, rock deformation and pore fluid flow involved in
the model development. Section 3.3 describes the geometries, boundary conditions and
materials of the finite-element framework for simulating both static and dynamic fluid
losses without and with drilling mud circulation in the well. Finally, in Section 3.4, the
simulation results are presented, and factors that influence static and/or dynamic fluid loss
are investigated and discussed based on the results.
3.2 NUMERICAL METHOD AND GOVERNING EQUATIONS
Lost circulation while drilling is simulated using a coupled fluid flow and
mechanics numerical model in this study based on the finite-element method. A lost
circulation system generally consists of three components: the well, the fracture, and the
formation. Figure 3.1 shows a typical configuration of the lost circulation system. The
following physical processes that happen in a lost circulation event are included and
simulated simultaneously in the proposed numerical model:
(1) Fluid circulation in the well.
(2) Lost circulation fracture propagation and fluid flow in the fracture.
56
(3) Formation rock deformation and pore fluid flow.
Figure 3.1: Illustration of lost circulation system with well, formation and fracture.
In this study, Abaqus®, a general purpose finite-element-method code for solving
linear and non-linear stress-analysis problems, is used for simulating the above physical
processes in the lost circulation problem.
3.2.1 Fluid Circulation in the Well
Fluid circulation in the well is modeled based on Bernoulli’s equation considering
gravity and viscous pressure losses. Flow between two points in a flow pipe, as shown in
Figure 3.2, is modeled based on Bernoulli’s equation as:
57
∆𝑃 − 𝜌𝑔∆𝑍 = 𝐶𝐿𝜌𝑣2
2 (3.1)
𝐶𝐿 =𝑓𝐿
𝐷ℎ (3.2)
where, ∆𝑃 is the pressure difference between the two points; ∆𝑍 is the elevation
difference between the two points; 𝑣 is the fluid velocity in the pipe; 𝜌 is the fluid
density; 𝑔 is the gravity acceleration factor; 𝐶𝐿 is the loss coefficient; 𝐿 is the pipe
length; 𝑓 is the friction factor; 𝐷ℎ is the hydraulic diameter of the pipe, which is a
function of the cross-section area 𝐴 and wetted perimeter 𝑆 of the pipe and expressed as
𝐷ℎ =4𝐴
𝑃.
The friction factor 𝑓 in Eq. 3.2 is an important parameter controlling the friction
loss of fluid flow. In the simulation, the friction factor can be determined by two methods.
The first method is using the Blasius friction loss formula which uses an empirical relation
based on Reynold’s number 𝑅𝑒 to determine friction factor (Hager, 2003; SIMULIA,
2016). This method distinguishes two different flow regimes according to Reynold’s
number, i.e. laminar flow when 𝑅𝑒 < 2500 and turbulent flow when 𝑅𝑒 > 2500. The
friction factors for the two different flow regimes are expressed as
{𝑓 =
64
𝑅𝑒 (𝑅𝑒 < 2500)
𝑓 =0.3164
𝑅𝑒0.25 (𝑅𝑒 ≥ 2500) (3.3)
The second method to determine the friction factor 𝑓 is based on the Churchill’s
formula which takes into account both the flow regimes (Reynold’s number) and the
roughness of the pipe (Churchill, 1977; SIMULIA, 2016). The friction factor is expressed
as
𝑓 = 8 [(8
𝑅𝑒)
12
+1
(𝐴+𝐵)1.5]
1
12
(3.4)
where,
𝐴 = [−2.457𝑙𝑛 ((7
𝑅𝑒)
0.9
+ 0.27𝐾𝑠
𝐷ℎ)]
16
;
58
𝐵 = (37350
𝑅𝑒)
16
;
and 𝐾𝑠 is pipe roughness.
It should be noted that there is a discontinuous jump in the friction factor when the
flow transitions from laminar to turbulent regime at 𝑅𝑒 = 2500 in the Blasius expression
(Eq. 3.3). This discontinuity may cause convergence issues in the simulation. However, the
Churchill’s formula (Eq. 3.4) transitions smoothly from laminar to turbulent flow. Figure
3.3 shows the transition behaviors of the two models for 𝐾𝑠 𝐷ℎ⁄ = 10−5. In the simulation
studies in this research, Churchill’s formula is used to model the friction loss behavior of
fluid flow in well. Pipe element in Abaqus® is used to represent the well in the model.
Figure 3.2: Schematic of fluid flow in pipe.
59
Figure 3.3: Friction factor determined from Blasius model and Churchill model: Blasius
model shows discontinuous transition from laminar to turbulent flow at
Re=2500, while Churchill model shows smooth transition.
3.2.2 Fracture Propagation and Fluid Flow in the Fracture
Fracture propagation and fluid flow in the fracture are modeled based on a cohesive
zone method using coupled pore pressure and deformation cohesive elements in Abaqus®.
A traction-separation constitutive law for describing fracture propagation and a fluid flow
constitutive law for describing fracture fluid flow are incorporated into the cohesive zone
model.
Fracture opening and propagation are modeled as the damage evolution between
two initially bonded interfaces with zero interfacial thickness. The traction-separation
constitutive law consists of three components: initial (before damage) loading behavior,
damage initiation, and damage evolution of the cohesive interface. Figure 3.4 shows the
traction-separation constitutive law used in the study. The initial loading process is
assumed to follow linear elastic behavior, determined by the stiffness of the interface which
relates stress and strain across the interface. Before damage initiation, the stiffness of the
interface remains constant. Damage begins when the stress/traction applied on the interface
60
satisfies certain damage initiation criteria. In this study, a maximum nominal stress
criterion is used to model damage initiation, which assumes that damage begins when the
traction on the interface reaches the tensile or shear strength (𝑇𝑜 in Figure 3.4) of the
interface.
Beyond damage initiation, the stiffness of the interface drops and damage evolution
occurs. Damage evaluation basically describes the rate at which the stiffness is degraded
once damage initiation is reached. There are two methods to define damage evaluation
either based on the relationship between the displacement at final failure 𝛿𝑓𝑖𝑛𝑎𝑙 (interface
stiffness drops to zero) and the displacement at damage initiation 𝛿𝑖𝑛𝑖 (interface stiffness
begins to drop), or based on the energy 𝐺𝐶 dissipated due to failure (SIMULIA, 2016). In
this study, damage evolution is defined based on an energy criterion called the
Benzeggagh-Kenane fracture criterion (Benzeggagh and Kenane, 1996):
𝐺𝑛𝐶 + (𝐺𝑠
𝐶 − 𝐺𝑛𝐶) (
𝐺𝑆
𝐺𝑇)
𝛽
= 𝐺𝐶 (3.5)
where, 𝐺𝑆 = 𝐺𝑠 + 𝐺𝑡 is the total energy dissipated due to deformations in the first and
second shear directions; 𝐺𝑇 = 𝐺𝑛 + 𝐺𝑠 + 𝐺𝑡 is the total energy dissipated due to
deformations in the normal, the first shear and the second shear directions; 𝐺𝐶 = 𝐺𝑛𝐶 +
𝐺𝑠𝐶 + 𝐺𝑡
𝐶 is the total critical fracture energy in the normal, the first shear and the second
shear directions; 𝐺𝑛, 𝐺𝑠 and 𝐺𝑡 are the energies dissipated due to deformations in the
normal, the first shear, and the second shear directions, respectively; 𝐺𝑛𝐶, 𝐺𝑠
𝐶 and 𝐺𝑡𝐶 are
the critical energies required to case failure in the normal, the first shear, and the second
shear directions, respectively. This fracture criterion assumes that when the ratio of energy
dissipated due to shear deformations to the total energy dissipated in the damage process
reaches a critical value determined by the critical fracture energies of the material, the
fracture will begin to propagate. This criterion is more appropriate for situations where the
61
critical fracture energies purely along the first and second shear direction are similar
(Wang, 2015; Yao et al., 2010). It is very suitable for modeling failure in geomaterials.
Figure 3.4: A typical traction-separation law
For modeling hydraulic driven fracture using the cohesive element method, upon
complete failure of the pore pressure cohesive element, i.e. the separation of the interface
reaching the critical value 𝛿𝑓𝑖𝑛𝑎𝑙 and stiffness of the element reducing to zero, fluid will
flow into the element. Fluid flow in the fracture includes two components as shown in
Figure 3.5: longitudinal flow along the fracture and normal fluid flow (leak-off) from
fracture faces to the surrounding porous medium. The mass conservation of the fluid inside
the fracture is governed by Reynold’s lubrication theory and can be expressed by the
continuity equation (Zielonka et al., 2014):
𝜕𝑤
𝜕𝑡+
𝜕𝑞𝑓
𝜕𝑠+ 𝑣𝑡 + 𝑣𝑏 = 0 (3.6)
where 𝑤 is the fracture aperture; 𝑞𝑓 is the longitudinal fluid flow rate in the fracture; 𝑣𝑡
and 𝑣𝑏 are the normal flow velocities through the top and bottom faces of the fracture,
which can be interpreted as fluid leak-off rate from the fracture to the surrounding porous
medium (Wang, 2015; Zielonka et al., 2014).
62
Incompressible and Newtonian fracturing fluid is assumed in this study. The
momentum equation of the tangential flow in the fracture can be expressed as that of a
Newtonian fluid flow between narrow parallel plates:
𝑞𝑓 = −𝑤3
12𝜇𝑓
𝜕𝑝𝑓
𝜕𝑠 (3.7)
where 𝜇𝑓 is the fluid viscosity, 𝑝𝑓 is the fluid pressure inside the fracture.
Fluid leak-off rates or normal fluid velocities are computed as:
𝑣𝑡 = 𝑐𝑡(𝑝𝑓 − 𝑝𝑡) (3.8)
𝑣𝑏 = 𝑐𝑏(𝑝𝑓 − 𝑝𝑏) (3.9)
where 𝑝𝑡 and 𝑝𝑏 are the pore fluid pressure in the porous medium adjacent to the top and
bottom faces of the fracture; 𝑐𝑡 and 𝑐𝑏 are the parameters control the fluid flow across
the top and bottom fracture faces, which is usually referred as “leak-off coefficients”. This
normal flow model can be interpreted as a thin layer of filter cake on the fracture faces,
which increases or reduces effective permeability of the fracture faces (Yao et al., 2010;
SIMULIA, 2016).
Figure 3.5: Schematic of fluid flow in the cohesive fracture (Modified after Zielonka et al.,
2014)
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3.2.3 Formation Rock Deformation and Pore Fluid Flow
The deformation of the porous formation and pore fluid flow in it are modeled using
coupled pore pressure and deformation continuum finite elements. The formation rock is
assumed to be an isotropic and poroelastic medium. With the sign convention used in the
finite-element analysis throughout this dissertation that tension is positive and compression
is negative, the relationship between total stress 𝝈, effective stress 𝝈′ , and pore pressure
𝑝𝑝, can be expressed as (Biot, 1941):
𝝈 = 𝝈′ − 𝛼𝑝𝑝𝑰 (3.10)
where 𝑰 is unit matrix; 𝛼 is the Biot coefficient, which is a material property of the porous
medium.
Stress equilibrium for the solid phase of the porous material is expressed using the
principle of virtual work for the volume under its current configuration (SIMULIA, 2016;
Wang, 2015; Yao et al., 2010):
∫ 𝝈 .. 𝛿𝜺 𝑑𝑉 = ∫ 𝒕 ∙ 𝛿𝒗 𝑑𝑆 + ∫ 𝒇 ∙ 𝛿𝒗 𝑑𝑉
𝑉
𝑆
𝑉 (3.11)
where 𝑉 is the control volume; 𝑆𝜎 is the surface area under surface traction; 𝝈 is the
total stress matrix, 𝛿�̇� is the virtual strain rate matrix; 𝒕 is the surface traction vector; 𝒇
is the body force vector; and 𝛿𝒗 is the virtual velocity vector. This equation is discretized
using a Lagrangian formulation for the solid phase, with displacements as the nodal
variables. The porous medium is thus modeled by attaching the finite element mesh to the
solid phase. Fluid is allowed to flow through these meshes.
Fluid flow should satisfy the continuity equation, which equates the rate of increase
in fluid volume stored at a point to the rate of volume of fluid flowing into the point within
the time increment:
𝑑
𝑑𝑡(∫ 𝜌𝑓
𝑉𝜑𝑑𝑉) = − ∫ 𝜌𝑓𝒏 ∙ 𝒗𝒇𝒑𝑑𝑆
𝑆 (3.12)
64
where 𝜌𝑓 is the density of the pore fluid; 𝜑 is the porosity of the medium; 𝒗𝒇𝒑 is the
average velocity of the pore fluid relative to the solid phase; 𝒏 is the outward normal to
surface 𝑆 . The continuity equation is integrated in time using the backward Euler
approximation and discretized with finite elements using pore pressure as the variable.
The pore fluid flow behavior in the formation in this study is assumed to be
governed by Darcy's law. It can be describe as:
𝒗𝒇𝒑 = −1
𝜑𝑔𝜌𝑓𝒌 ∙ (
𝜕𝑝𝑝
𝜕𝑿− 𝜌𝑓𝒈) (3.13)
where 𝒈 is the gravity acceleration vector; 𝑔 is the gravity acceleration magnitude; 𝒌
is the hydraulic conductivity of the porous medium; 𝑝𝑝 is pore pressure; 𝑿 is a spatial
coordinate vector. As can be seen from Eqs. 3.10 through 3.13, the stress and pore fluid
pressure in the porous medium are nonlinearly coupled with each other to form a control
equation. When it is converted into a weak form of the equivalent integral, it can be solved
by the finite element discretization method.
3.3 LOST CIRCULATION MODEL
This section describes the assumptions, geometries, boundary conditions, and
materials of two finite-element models developed for simulation of static and dynamic fluid
losses without and with circulation of drilling fluid.
3.3.1 Static Fluid Loss Model
Lost circulation in vertical wellbore is simulated in this study as illustrated in Figure
3.1. While drilling, the drilling fluid is pumped into the well through drilling pipe in the
center of the wellbore and returns to the surface through the annulus between the drilling
pipe and wellbore wall. In static state, pump is stopped. Therefore, there is no fluid flow in
the wellbore. The bottom hole pressure (BHP) is solely dependent on the gravity of the
fluid column in the wellbore.
65
3.3.1.1 Model Geometry
The well and formation are assumed to be in a plane-strain condition at a certain
depth, therefore a 2D geometry is used for the model, as shown in Figure 3.6. Owing to
symmetry, only one half of the formation is considered as shown in Figure 3.7.
The size of the modeled formation is 60×20 m. The well depth is 1000 m. The
overburden stress is in the Z-direction perpendicular to the horizontal plane, and the
maximum and minimum horizontal stresses are in the X- and Y- direction respectively in
the horizontal plane. A predefined fracture path is assigned in the middle of the model
perpendicular to the direction of the minimum horizontal stress. The formation is modeled
as an isotropic, poroelastic material, using coupled pore pressure and deformation
continuum finite elements; the fracture is modeled using a layer of pore pressure cohesive
elements; and the well is model with pipe elements. Since significant stress/displacement
gradients are expected in the wellbore vicinity, the mesh is refined around the wellbore.
Figure 3.6: Schematic configuration of well and formation. The formation is in a plane-
strain condition.
66
Figure 3.7: Static lost circulation model.
3.3.1.2 Boundary Conditions
A symmetric boundary condition is defined on the left edge of the model as shown
in Figure 3.7. The formation is assumed to be at a depth of 1000 m with a normal pore
pressure of 10 MPa, a minimum horizontal stress of 13 MPa and a maximum horizontal
stress of 15 MPa. The minimum and maximum horizontal stresses and undisturbed pore
pressure (10 MPa) are applied on the outer boundaries of the model, as shown in Figure
3.7. Initial pore pressure of 10 MPa is applied to the whole formation.
It is also assumed that the well is always filled up with drilling fluid even after fluid
loss occurs. Gravity force is applied to the fluid in the wellbore. The pressure at the
wellhead is equal to atmospheric pressure (assumed to be zero in this study since its small
value compared with fluid pressure in the wellbore).
The pipe element at the end of the wellbore is tied to the formation and fracture
elements on the wellbore wall to make sure the fluid pressure in the bottom hole is equal
67
to the pore pressure on the wellbore wall. A dynamic pressure that equals the bottom-hole
fluid pressure is imposed onto the inner wellbore wall to model the pushing pressure
applied by fluid column. Besides, the tie constraint enforces the fluid conservation and
equilibrium between the wellbore and formation before facture initiation. When fracture
initiates, the tie constraint handles the fluid conservation and equilibrium between the
wellbore, the formation, and the fracture.
3.3.1.3 Material Properties
The formation rock is modeled as a poroelastic material. It is assumed to be a
sandstone formation with a high permeability of 100 mD. Drilling fluids with different
densities are used to investigate its effect on static fluid loss. Table 3.1 summarizes all the
material parameters used for the simulations.
Table 3.1: Material properties of the static fluid loss model.
Parameters Values Units
Young’s modulus 7000 MPa
Poisson’s ratio 0.2
Fluid density 1.0, 1.2, 1.4, 1.5, 1.6 g/cm3
Fluid viscosity 1 cp
Porosity 0.25
Permeability 100 mD
Tensile strength 0.4 MPa
Critical fracture energy 28 J/m2
3.3.2 Dynamic Fluid Loss Model
The objective of the dynamic fluid loss model is to model lost circulation in vertical
wellbores with drilling fluid circulation. While drilling, the drilling fluid is pumped into
the well through drilling pipe in the center of the wellbore and returns to the ground through
the annulus between the drilling pipe and well wall. The BHP or ECD during this process
68
is dependent not only on the gravity of the fluid column but also on the viscous loss in the
annulus.
3.3.2.1 Model Geometry
The geometry of the dynamic fluid loss model is very similar to the static fluid loss
model as describe in Section 3.3.1.1. The only difference is that the well is modeled as a
“U-tube” configuration, rather than as a single tube as defined in the static model. The
drilling pipe is modeled using fluid pipe connector element in Abaqus® in order to simulate
fluid pumping into and flow through it. The wellbore annulus is modeled using fluid pipe
element in Abaqus®. And they are connected to each other at the bottom of the wellbore as
shown in Figure 3.8.
Figure 3.8: Dynamic fluid loss model.
69
3.3.2.2 Boundary Conditions
It is assumed that the drilling pipe and annulus are filled up with drilling fluid
during circulation. The pressure at the annulus head is equal to atmospheric pressure
(assumed to be zero in this study due to its small magnitude compared with fluid pressure
in the wellbore and formation). A fluid pumping rate is applied at the top of drilling pipe.
The drilling pipe is connected to the annulus at the bottom of the wellbore. Since the
drilling fluid is assumed to be incompressible, the fluid flow rate into the annulus at the
wellbore bottom is equal to the pump rate. Because BHP is only controlled by the fluid in
the annulus, therefore fluid gravity and viscous pressure loss are only considered for this
part of fluid. The other boundary conditions are the same as those defined in the static fluid
loss model in Section 3.3.1.2.
3.3.2.3 Input Parameters
As aforementioned, BHP or ECD is affected by the gravity pressure of the fluid as
well as the viscous pressure loss due to fluid flow in the annulus. At a certain depth, the
gravity pressure of the fluid is only determined by the density of drilling mud, while the
viscous pressure loss is influenced by several factors, including mud density, mud
viscosity, flow (pump) rate, and the size/clearance of the annulus. In order to investigate
the effects of these factors on lost circulation, different fluid properties, pump rates and
annulus clearances are used in this study as summarized in Table 3.2. The other parameters
used in the simulations are the same as those given in Section 3.3.1.3.
70
Table 3.2: Input parameters for the dynamic fluid loss model.
Parameters Values Units
Mud Density 1.0, 1.1*, 1.2, 1.3, 1.35, 1.4, 1.5 g/cm3
Mud Viscosity 1*, 10, 50, 100, 500 cp
Pump Rate 0.36*, 0.42, 0.48, 0.54, 0.60 m3/min
Wel
l C
onfi
gura
tion Drilling Pipe
Radius 5* 5 5 5 cm
Wellbore Radius 10* 9 8 7 cm
Annulus Clearance 5* 4 3 2 cm
Cross-section Area 235.6* 175.9 122.5 75.3 cm2
Hydraulic Diameter 10* 8 6 4 cm
Note: Values with * are the inputs for the base case of the simulations discussed in the flowing
section.
3.4 FLUID LOSS SIMULATION RESULTS
Using the lost circulation simulation framework described above, fluid losses in
static and dynamic states without and with well circulation are simulated and analyzed
under various conditions. This section summarizes the analysis results.
3.4.1 Static Fluid Loss
For a formation at a certain depth, BHP and thus lost circulation are only dependent
on the drilling mud density under static state with mud circulation. Therefore, different
mud densities are used in the simulations to investigate its effect on static fluid loss.
Figures 3.9 and 3.10 show the fluid loss rates and BHP with different drilling mud
densities. The following conclusions are made based on the simulation results.
With a mud density of 1.0 𝑔 𝑐𝑚3⁄ , the BHP is equal to the pore pressure (10 MPa)
in the formation (see Figure 3.10), therefore no fluid loss occurs, i.e. the fluid loss
rate is zero as shown in Figure 3.9.
71
When the mud density increases to 1.2 or 1.4 𝑔 𝑐𝑚3⁄ , the BHP becomes larger
than pore pressure and there is a very small fluid loss rate. The simulation results
do not show occurrence of fractures for this two cases, therefore this small amount
of fluid loss is due to fluid seepage from wellbore wall to the formation, rather than
fluid flow into fractures.
When the mud density is equal or higher than 1.5 𝑔 𝑐𝑚3⁄ , fracture occurs. It can
be seen that there is a rapid increase in the loss rate at the early time due to the
initiation and propagation of the fractures. But finally it approaches a relatively
constant value. As expected, it is observed from Figure 3.9 that the larger the mud
density, the higher the fluid loss rate. Figure 3.10 shows that the BHP declines
rapidly with the creation of the fractures and finally approaches a constant fracture
propagation pressure which is even smaller than the BHP with 1.4 𝑔 𝑐𝑚3⁄ mud.
It is also observed that the larger the mud density, the faster the BHP drops,
meaning the faster of the fracture creation at the early time.
72
Figure 3.9: Fluid loss rate with different mud densities in the static fluid loss model.
Figure 3.10: BHP with different mud densities in the static fluid loss model.
73
Figure 3.11 displays the lost circulation fracture geometry with different mud
densities at the end of the simulation (100s). With a mud density of 1.4 𝑔 𝑐𝑚3⁄ , no fracture
happens. And then, increasing the mud density by 0.1 𝑔 𝑐𝑚3⁄ results in significant
fracture growth. With the continuous increase of mud density, both of the fracture width
and fracture length increase.
Figure 3.11: Lost circulation fracture geometry with different mud densities in the static
fluid loss model.
3.4.2 Dynamic Fluid Loss
The numerical model framework developed in this chapter allows for a unique new
way of modeling fractures while drilling fluid circulation as “dynamic-pressure-driven”
fractures. Given the conditions of formation depth, rock properties, wellbore
configurations, mud properties and pump schedules, the model can capture the dynamic
loss rate, reduction of return circulation, BHP, and fracture geometry with dynamic fracture
growth. The analysis of the fluid loss and fracture propagation using the simulation
74
framework can help us understand how to prevent lost circulation, optimize mud rheology,
and select LCMs.
3.4.2.1 Effect of Mud Density
Figures 3.12, 3.13 and 3.14 show return circulation rate, fluid loss rate and BHP
while circulating drilling fluids with different densities. The following observations can be
reached from the simulation results.
With a mud density of 1.0 𝑔 𝑐𝑚3⁄ , while the BHP is equal to formation pressure
(10 MPa) without fluid circulation in the static state as shown in Figure 3.10, it is
not equal to formation pressure anymore in the dynamic model with fluid
circulation as shown in Figure 3.14. The BHP with fluid circulation is about 2.2
MPa higher than the hydrostatic formation pressure. This increase of BHP is caused
by the viscous loss due to fluid flow in the annulus. No fracture occurs in this case.
However, because of the pressure difference between BHP and formation pore
pressure, a slow fluid loss occurs as evidenced by the slight reduction in the return
circulation rate and the small fluid loss rate in Figures 3.12 and 3.13 respectively.
When the mud density increases to 1.2 𝑔 𝑐𝑚3⁄ , the BHP becomes larger. But no
fracture occurrence is observed. There is a small amount of fluid loss due to seepage
through wellbore wall.
With a mud density of 1.3 𝑔 𝑐𝑚3⁄ , fracture occurs with a dramatic reduction in the
return circulation rate and increase in the fluid loss rate. The BHP also drops with
fracture creation. Partial loss occurs in this case since the fluid loss rate is smaller
than the pump rate. Recall that for the static model described in the above section,
fracture does not occur until the mud density is increased to 1.5 𝑔 𝑐𝑚3⁄ .
75
With continuous increase of mud density to 1.35, 1.4 and 1.5 𝑔 𝑐𝑚3⁄ , total losses
are observed. It can be seen from Figures 3.12 and 3.13 that the return circulation
rates drop to negative values and the loss rates become larger than the pump rate,
which means no fluid returns to the ground surface through the annulus, rather the
fluid in the annulus will flow backward into the fractures. Since the annulus is
assumed always filled with fluid, the decline of fluid level in the annulus is not
considered. As expected, the larger the fluid density, the larger the fluid loss rate is
observed. Figure 3.14 shows that the BHP declines rapidly with the creation of the
fractures and finally approaches a constant fracture propagation pressure of 13.5
MPa.
Figure 3.12: Return circulation rate with different mud densities in the dynamic fluid loss
model.
76
Figure 3.13: Fluid loss rate with different mud densities in the dynamic fluid loss model.
Figure 3.14: BHP with different mud densities in the dynamic fluid loss model.
77
Figure 3.15 displays the fracture geometry with different mud densities after a
circulation period of 100s. With a mud density of 1.2 𝑔 𝑐𝑚3⁄ , no fracturing happens.
Increasing the mud density to 1.3 𝑔 𝑐𝑚3⁄ results in significant fracture growth. However,
in the static model without fluid circulation, fracture begins to occur at a much higher mud
density of 1.5 𝑔 𝑐𝑚3⁄ . With the continuous increase of mud density, both of the fracture
width and fracture length increase.
Figure 3.15: Lost circulation fracture geometry with different mud densities in the dynamic
fluid loss model.
Figure 3.16 compares the fracture geometry in the static loss case and dynamic loss
case with the same mud density of 1.5 𝑔 𝑐𝑚3⁄ at the same time of 100s. It is obvious that
considering fluid circulation and the corresponding viscous loss in the annulus can result
in significantly larger fracture size, even though the mud density is the same. Therefore, it
is important to take into account the dynamic circulation effect on lost circulation
prediction and prevention.
78
Figure 3.16: Comparison of fracture geometry between the static and dynamic fluid loss
models.
3.4.2.2 Effect of Mud Viscosity
As described in Section 3.2, drilling mud viscosity is one of the dominating factors
for viscous pressure loss. Therefore, it can influence BHP and ECD, and hence lost
circulation, while drilling. In this section, dynamic fluid loss and fracture geometry for
mud with different viscosities are analyzed. A constant mud density of 1.1 𝑔 𝑐𝑚3⁄ is used
in this section.
Figures 3.17, 3.18 and 3.19 show return circulation rate, fluid loss rate and BHP
while circulating drilling fluids with different viscosities.
With a mud viscosity of 1cp, the dynamic BHP with fluid circulation is about 2.3
MPa higher than the static BHP (11 MPa with a mud density of 1.1 𝑔 𝑐𝑚3⁄ ) as
shown in Figure 3.18. This increase of BHP is caused by the viscous pressure loss
in the annulus. When the mud viscosity is increased to 10cp, the BHP increases to
about 3.2 MPa above the static BHP due to the increased viscous loss. No fracture
79
occurs in this two cases. However, because of the pressure difference between BHP
and formation pore pressure, a slow fluid loss occurs as evidenced by the slight
reduction in the return circulation rate and the small flow loss rate in Figures 3.17
and 3.18 respectively.
For the cases with a mud viscosity of 50, 100 and 500 cp, fracture occurs with a
dramatic reduction in the return circulation rate and increase in the fluid loss rate
as shown in Figures 3.17 and 3.18. Figure 3.19 shows that the BHP declines rapidly
with the creation of the fracture and finally approaches a constant fracture
propagation pressure of 13.5 MPa. Partial losses occur in these cases since the fluid
loss rate is smaller than the pump rate. Fluid loss rate increases with the increase of
mud viscosity due to the larger viscous loss in the annulus and thus higher ECD at
the bottom hole.
Figure 3.17: Return circulation rate with different mud viscosities in the dynamic fluid
loss model.
80
Figure 3.18: Fluid loss rate with different mud viscosities in the dynamic fluid loss
model.
Figure 3.19: BHP with different mud viscosities in the dynamic fluid loss model.
81
Figure 3.20 displays the fracture geometry with different mud viscosities after a
circulation period of 100s. With a mud viscosity of 10 cp, no fracturing occurs. Increasing
the mud viscosity to 50 cp results in clear fracture growth. With the continuous increase of
mud viscosity, both of the fracture width and fracture length increase due to the increased
ECD in the bottom annulus.
Figure 3.20: Lost circulation fracture geometry with different mud viscosities in the
dynamic fluid loss model.
3.4.2.3 Effect of Pump Rate
Pump rate dominates the fluid flow velocity in the annulus which is another control
factor for viscous loss. Increasing pump rate will lead to increased BHP or ECD in the
annulus associated with the increase of viscous loss. In this section, the dynamic fluid loss
and fracture geometry with different pump rates are analyzed.
Figures 3.21, 3.22 and 3.23 show return circulation rate, fluid loss rate and BHP
while circulation with different pump rates.
82
With a pump rate of 0.36 𝑚3 𝑚𝑖𝑛⁄ , the BHP is around 13.3 MPa which is 2.3 MPa
higher than the static BHP as shown in Figure 3.23. This increase of BHP is caused
by the flow viscous loss in the annulus. When the pump rate is increased to 0.42
𝑚3 𝑚𝑖𝑛⁄ , the BHP increases to 14.2 MPa due to the increased viscous loss. No
fracture occurs in this two cases. However, because of the pressure difference
between BHP and formation pressure, a slow fluid loss occurs as evidenced by the
slight reduction in the return circulation rate and the small flow loss rate in Figures
3.21 and 3.22 respectively.
For the cases with pump rates of 0.48, 0.54 and 0.60 𝑚3 𝑚𝑖𝑛⁄ , fracture occurs with
a dramatic reduction in the return circulation rate and increase in the fluid loss rate
as shown in Figures 3.21 and 3.22. Partial fluid losses occur in these cases. The
return circulation rates approach to a similar value as fracture grow for this three
cases with different pump rates. However, the loss rate increases as the pump rate
increases. Figure 3.23 shows that the BHP declines rapidly with the creation of the
fractures and finally approaches a constant fracture propagation pressure of 13.5
MPa for this three cases.
83
Figure 3.21: Return circulation rate with different pump rates in the dynamic fluid loss
model.
Figure 3.22: Fluid loss rate with different pump rates in the dynamic fluid loss model.
84
Figure 3.23: BHP with different pump rates in the dynamic fluid loss model.
Figure 3.24 displays the fracture geometry with different pump rates at the end of
a circulation period of 100s. With a pump rate of 0.42 𝑚3 𝑚𝑖𝑛⁄ , no fracture occurs.
Increasing the pump rate to 0.48 𝑚3 𝑚𝑖𝑛⁄ results in significant fracture growth. With the
continuous increase of pump rate, both of the fracture width and fracture length increase
due to the increase of viscous loss in the annulus and ECD in the bottom hole.
85
Figure 3.24: Lost circulation fracture geometry with different pump rates in the dynamic
fluid loss model.
3.4.2.4 Effect of Annulus Clearance
Changing the annulus clearance between the drilling pipe and wellbore wall lead to
changes in flow velocity and hydraulic diameter of the flow conduit, which can further
influence the viscous loss as described in Section 3.2. Generally, smaller annulus clearance
has larger flow velocity and smaller hydraulic diameter, and therefore results in larger
viscous loss according to Eqs. 3.1 and 3.2. In this section, dynamic fluid loss and fracture
geometry with different annulus clearances as given in Table 3.2 are analyzed based on the
simulation results.
Figures 3.25, 3.26 and 3.27 show return circulation rate, fluid loss rate and BHP
while circulation with different annulus clearances. The other input parameters are the
same as those given in Table 3.2.
With a relatively larger annulus clearance of 5cm between the drilling pipe and
wellbore wall, no fracture occurs. However, because of the pressure difference
86
between BHP and formation pressure, a slow fluid loss occurs as evidenced by the
slight reduction in the return circulation rate and the small flow loss rate in Figures
3.25 and 3.26 respectively.
For the cases with an annulus clearance of 4, 3 and 2 cm, fracture occurs due to the
increased viscous loss or ECD in the annulus. Significant reduction in the return
circulation rate and increase in the fluid loss rate are observed as shown in Figures
3.25 and 3.26. Partial losses occurs in these cases. The smaller the annulus
clearance, the smaller the return circulation rate and the larger the fluid loss rate.
Figure 3.27 shows that the BHP declines rapidly with the creation of the fractures
and finally approaches a constant fracture propagation pressure of 13.5 MPa for
this three cases.
Figure 3.25: Return circulation rate with different annulus clearances in the dynamic fluid
loss model.
87
Figure 3.26: Fluid loss rate with different annulus clearances in the dynamic fluid loss
model.
Figure 3.27: BHP with different annulus clearances in the dynamic fluid loss model.
88
Figure 3.28 displays the fracture geometry with different annulus clearances at the
end of a circulation period of 100s. With a relative larger annulus clearance of 5 cm, no
fracturing occurs. Decreasing the annulus clearance to 4 cm results in significant fracture
growth. With the continuous decrease of annulus clearance, both of the fracture width and
fracture length increase due to the increase of viscous loss in the annulus.
Figure 3.28: Lost circulation fracture geometry with different annulus clearances in the
dynamic fluid loss model.
3.5 SUMMARY
In this chapter, a finite-element framework is developed which allows predicting
the dynamic fluid loss and fracture geometry during lost circulation in drilling process. It
can be used to simulate fluid loss in static and dynamic state without and with drilling fluid
circulation in the wellbore. The model successfully couples the fluid circulation in the
wellbore, fluid seepage on wellbore wall, fracture propagation, fluid flow in fracture, pore
fluid flow and deformation of formation rock during fluid loss through induced fractures.
89
The fluid flow in the wellbore is modeled based on the Bernoulli’s equation taking into
account the viscos loss. Fracture propagation and fluid flow in the fracture are modeled
based on a pore pressure cohesive zone method.
Factors that can affect ECD and thus lost circulation, including mud density, mud
viscosity, pump rate and annulus clearance, are investigated using the proposed model. The
results show that the viscous pressure loss term due to fluid circulation in the annulus can
lead to significant ECD increase and fluid loss. Drilling mud with relatively low density
which does not cause lost circulation in static state without circulation may lead to
significant fluid loss after resuming mud circulation. So it is important to take into account
the dynamic circulation effect on lost circulation prediction and prevention.
The numerical framework provides a unique new way to model lost circulation in
drilling when the boundary condition at the fracture mouth is neither a constant flowrate
into the fracture nor a constant pressure, but rather a dynamic BHP. In drilling operations,
we are interested in preventing fractures from occurring by controlling BHP/ECD, or
plugging the fractures at the early time of their growth using LCMs, so the capability of
capturing the dynamic fluid loss and fracture geometry development of the proposed
framework can help us understand how to prevent lost circulation, optimize mud rheology,
and select LCMs.
90
CHAPTER 4: Modeling Study of Preventive Wellbore Strengthening
Treatments: The Role of Mudcake
Preventive wellbore strengthening treatment is to add additives, such as LCMs, to
drilling mud to build a layer of low-permeability mudcake on wellbore wall which can
inhibit fracture creation and increase the sustainable pressure of the wellbore. Mudcake
buildup on wellbore wall plays an important role on preventing lost circulation. However,
the time-dependent mudcake buildup and properties have been plaguing the drilling
industry for years. In this chapter, an analytical model is first derived to quantify the effects
of mudcake thickness, permeability and strength on wellbore stresses and fracture pressure.
Steady-state fluid flow is assumed for the analytical model, so it does not consider the time-
dependent effects. For a further step to take into account the time-dependent effects, a
finite-element model is developed to simulate the evolution of near-wellbore stresses and
pore pressure with dynamic mudcake buildup and property (permeability) change while
drilling based on poroelastic theory using Abaqus® and FORTRAN® subroutines.
91
4.1 INTRODUCTION
As the bit drills into a new permeable formation, fresh rock surface is exposed to
drilling mud. The wellbore pressure is usually higher than pore pressure in the surrounding
formation. Under the pressure difference, liquid component in the drilling mud may
seepage into the formation, while some solid mud components may be filtered out onto the
wellbore surface in this process. The filtered solid components can form a thin layer of
low-permeability cake on wellbore wall which is usually called mudcake or filtercake. The
buildup and properties of the mudcake depend on a number of factors such as drilling mud
constituent, wellbore pressure and temperature, pore pressure, formation permeability and
porosity, annulus flow regimes, and time.
Extensive experimental work in the literature has revealed the importance of
the mudcake in inhibiting fracture growth and preventing lost circulation (Cook et al.,
2016; Salehi et al., 2016; Salehi and Kiran, 2016). Researchers have found that adding
additives such as LCMs to drilling mud to build a layer of low-permeability mudcake can
enhance the effective strength of the wellbore (Song and Rojas, 2006; Soroush et al., 2006;
Sweatman et al., 2004). While some researchers insisted that wellbore strengthening is
achieved by bridging fractures on wellbore to increase the wellbore hoop stress (Alberty
and McLean, 2004; Aston et al., 2004b; Dupriest, 2005), other researchers have argued that
similar wellbore strengthening results can arise by build a low-permeability mudcake on
wellbore wall to alter the effective stresses on and around wellbore (Abousleiman et al.,
2007; Tran et al., 2011). Therefore, preventive wellbore strengthening technique based on
plastering the wellbore wall with mudcake before lost circulation occurs has been widely
used and proven to be very effective in the drilling industry, especially for preventing lost
circulation problems in depleted sandstone formations with relatively high permeability.
92
However, preventive wellbore strengthening treatment is a time-dependent
process, because not only the fluid flow in the porous rock around the wellbore is dependent
on time, but also the thickness and physical properties of mudcake are functions of time.
The time-dependent mudcake thickness and properties can significantly affect fluid flow
and therefore stress state around wellbore. This complex feature of mudcake has been
remaining a challenging problem in mudcake modeling in the drilling industry.
In this chapter, an analytical model assuming steady-state fluid flow around
wellbore and thus without considering the time-dependent effects is first derived to
quantify the effects of mudcake thickness, permeability, and strength on pore pressure and
effective stresses in the vicinity of the wellbore. Next, in order to take into account the
time-dependent effects, a finite-element model is developed to simulate the evolution of
near-wellbore stresses and pore pressure with time-dependent fluid flow and dynamic
mudcake thickness buildup and permeability reduction, based on poroelasticity theory
using Abaqus® and FORTRAN® subroutines.
4.2 AN ANALYTICAL MUDCAKE MODEL
4.2.1 Modeling Assumptions
In this section, an analytical mudcake model is derived which takes into account
the effects of thickness, permeability and strength of mudcake on near-wellbore pore
pressure and stress states, and thus fracture pressure of wellbore. Before deriving the
model, the following assumptions are introduced:
1. The wellbore, mudcake, and formation rock are in a plane-strain condition as
depicted in Figure 4.1.
2. The outer formation boundary has a constant pore fluid pressure 𝑃𝑒; the wellbore
is under a constant mud pressure 𝑃𝑖; and 𝑃𝑖 is higher than 𝑃𝑒.
93
3. The fluid flow from wellbore to formation is in steady state and obeys Darcy’s flow.
4. The formation rock is isotropic, homogeneous and poroelastic material.
5. The mudcake is very soft/flexible compared to the rock and has a very low yield
strength; therefore it undergoes perfectly plastic yielding under wellbore pressure;
its Poisson’s ratio for post-yield deformation is assumed to be 0.5.
6. Mudcake thickness and properties do not change with time.
7. The wellbore radius (outer mudcake radius) is 𝑅𝑜; the inner mudcake radius is 𝑅𝑖;
the outer formation radius is 𝑅𝑒 ; the mudcake thickness is 𝑤 = 𝑅𝑜 − 𝑅𝑖 ; the
mudcake permeability is 𝐾1; and the formation permeability is 𝐾2.
8. The maximum and minimum far-field total stresses are 𝜎𝐻 and 𝜎ℎ, respectively.
9. The sign convention for stress in this analytical study is that compression is positive
and tension is negative.
Figure 4.1: Schematic of the cross section of wellbore, mudcake, and formation.
94
4.2.2 Wellbore Stress and Fracture Pressure
Since there is fluid flow from the wellbore to formation, the pore pressure varies
with radial distance. In this problem, three components contributes to the stress around
wellbore: (1) wellbore pressure and field stresses; (2) varying pore pressure distribution
induced by fluid flow; and (3) the contribution of the plastic mudcake. For deriving the
solution, we assume these terms are uncoupled, so the final expressions can be obtained by
a superposition of the three problems. The following subsections describe the detailed
derivation process.
4.2.2.1 Total Stress Induced by Varying Pore Pressure
Assuming the pore pressure at wellbore wall (interface between mudcake and
formation) is 𝑃𝑜, then according to Darcy’s law for radial flow, one can have
1 22 2
ln lno i e o
o e
i o
K KP P P P
R R
R R
(4.1)
Solving this equation, one can get the pore pressure at the wellbore wall
2
1 2
ln
ln ln
o
io i i e
e o
o i
RK
RP P P P
R RK K
R R
(4.2)
Introduce
o i eP P P (4.3)
2
1 2
ln
ln ln
o
i
e o
o i
RK
RB
R RK K
R R
(4.4)
Eq. 4.2 can be simplified as
o i oP P B P (4.5)
95
Next, we want to determine the pore pressure in the formation around wellbore.
Define 𝑃𝑟 is the pore pressure in the formation at distance 𝑟 from wellbore center,
therefore 𝑅𝑜 ≤ 𝑟 ≤ 𝑅𝑒. The following equation can be written from Darcy’s law
2 22 2
ln lnr o e r
e
o
K KP P P P
r R
R r
(4.6)
Inserting the pore pressure at wellbore wall, i.e. Eq. 4.2, into Eq. 4.6, pore pressure at
distance 𝑟 can be obtained
2
1 2
ln ln1
ln
ln ln ln
o e
ir i o
e e oo
o o i
R RK
R rrP P P
R R RRK K
R R R
(4.7)
Define
2
1 2
ln ln1
ln
ln ln ln
o e
i
e e oo
o o i
R RK
R rrM
R R RRK K
R R R
(4.8)
Eq. 4.7 can be simplified to
r i oP P M P (4.9)
or
(1 )r e oP P M P (4.10)
Total stress induced by varying pore pressure around wellbore due to fluid flow can
be calculated by (Fjar et al., 2008)
2 2' ' ' ' ' '
, 2 2 2
2 2' ' ' 2 ' ' '
, 2 2 2
' ' '
z, 2 2
2
2
4
e
o o
e
o o
e
o
r Ro
r pR R
e o
r Ro
pR R
e o
R
pR
e o
r Rr P r dr r P r dr
r R R
r Rr P r dr r P r r P r dr
r R R
vr P r dr
R R
(4.11)
96
where, 𝜎𝑟,𝑝 , 𝜎𝜃,𝑝 and 𝜎𝑧,𝑝 are the total radial, tangential and axial stresses around
wellbore induced by varying pore pressure with fluid flow; ∆P(𝑟) = 𝑃𝑟 − 𝑃𝑒 ; 𝜂 is the
poroelastic stress coefficient; and 𝑣 is Poisson’s ratio. Using Eq. 4.7, ∆𝑃(𝑟) can be
expressed as
2
1 2
ln1
1 ln ln
ln ln ln
o
i eo
e e oo
o o i
RK
R RrP r P
R R RR rK K
R R R
(4.12)
Define
1
ln e
o
AR
R
(4.13)
and combine Eq. 4.4, ∆𝑃(𝑟) can be written as
1 ln ln eo
o
RrP r A AB P
R r
(4.14)
Let
' ' '
1o
r
RI r P r dr (4.15)
' ' '
2
e
o
R
RI r P r dr (4.16)
Inserting Eq. 4.14 into Eqs. 4.15 and 4.16, one can get '
' '
1 '1 ln ln
o
re
oR
o
RrI r A AB P dr
R r
(4.17)
'' '
2 '1 ln ln
e
o
Re
oR
o
RrI r A AB P dr
R r
(4.18)
After integration, 𝐼1 and 𝐼2 are obtained as
'2 ''2 '2
1 '
1 12ln 1 2ln 1
2 4 4oo o
rr r
eo
o RR R
Rr rI A r AB r P
R r
(4.19)
97
'2 ''2 '2
2 '
1 12ln 1 2ln 1
2 4 4
ee e
oo o
RR R
eo
o RR R
Rr rI A r AB r P
R r
(4.20)
Inserting Eqs. 4.19 and 4.20 into Eq. 4.11, total stresses around wellbore induced by
varying pore pressure caused by fluid flow can be determined as
2 2
, 1 22 2 2
2 22
, 1 22 2 2
z, 22 2
2
2
4
or p
e o
op
e o
p
e o
r RI I
r R R
r RI r P r I
r R R
vI
R R
(4.21)
4.2.2.2 Total Stress Induced by Wellbore Pressure, Far-field Stresses and Plastic
Mudcake
The total stress around wellbore induced by pressure at wellbore wall and far-field
stresses can be obtained using Kirsch Equation as
2 4 2 2
, 2 4 2 2
2 4 2
, 2 4 2
2
, 2
1 1 3 4 cos 22 2
1 1 3 cos 22 2
2 cos 2
H h o H h o o or s iw
H h o H h o os iw
oz s v H h
R R R RP
r r r r
R R RP
r r r
R
r
(4.22)
where 𝜎𝑟,𝑠 , 𝜎𝜃,𝑠 and 𝜎𝑧,𝑠 are the total radial, tangential and axial stresses around
wellbore induced by far-field stresses and pressure at wellbore wall; 𝜃 is the
circumferential angle to the direction of 𝜎𝐻. Note that in this equation, 𝑃𝑖𝑤 is the pressure
acting on wellbore wall; it is not equal to fluid pressure 𝑃𝑖 acting on the inner surface of
mudcake. 𝑃𝑖𝑤 is influenced by both the fluid flow through the mudcake and the plastic
flow of the mudcake.
From classical mechanics, the mudcake must satisfy the equilibrium condition
98
0rrd
dr r
(4.23)
The stress-strain relationship for the mudcake is
r r z
r z
z z r
E v
E v
E v
(4.24)
Due to its very soft/flexible feature, mudcake can be assumed to be in perfectly
plastic yielding condition with very low yield strength under the wellbore pressure (Tran
et al., 2011). The Poisson’s ratio for mudcake plastic flow is assumed to be 0.5. The
mudcake is considered in plane-strain condition, i.e. 휀𝑧 = 0. Then from the last equation
of Eq. 4.24, one can get
0.5z r (4.25)
According to the von Mises yield theory (also known as maximum distortion
energy theory), when mudcake yields, the following expression should be satisfied
(Aadnøy and Belayneh, 2004)
2 2 21
2r r z zY
(4.26)
where 𝑌 is the yield strength of the mudcake.
Inserting Eq. 4.25 into Eq. 4.26, one can get
3
2rY (4.27)
Inserting Eq. 4.27 into the equilibrium equation (Eq. 4.23), one obtains the radial stress
distribution
2
ln3
r
Yr C (4.28)
where C is an integration constant.
Boundary condition at the inner mudcake surface is
99
r i
i
P
r R
(4.29)
Applying this boundary condition into Eq. 4.28, the integration constant can be determined
as
2
ln3
i i
YC P R (4.30)
Inserting Eq. 4.30 into Eq. 4.28, radial stress distribution in the mudcake can be
determined as
2ln
3r i
i
Y rP
R
(4.31)
where 𝑅𝑖 ≤ 𝑟 ≤ 𝑅𝑜.
On the wellbore wall 𝑟 = 𝑅𝑜, the radial stress is
2( ) ln
3
or o i
i
RYR P
R
(4.32)
Because in most cases the mudcake is extremely soft/flexible compared to the rock
formations, it can be assumed that the mudcake does not exert any shear tractions on the
wellbore wall (Tran et al., 2011), but it can transmit the radial stress exerted by drilling
fluid pressure on the inner surface of mudcake. The radial stress on the inner surface of
mudcake is equal to drilling fluid pressure 𝑃𝑖, while the radial stress on the wellbore wall
is given by Eq. 4.32. By simply replacing 𝑃𝑖𝑤 in the Kirsch solution (Eq. 4.22) with
𝜎𝑟(𝑅𝑜) in Eq. 4.32, the total stress induced by wellbore pressure, far-field stress and
plasticity of mudcake can be determined.
4.2.2.3 Total Stress around Wellbore
Replacing 𝑃𝑖𝑤 in Kirsch solution (Eq. 4.22) with 𝜎𝑟(𝑅𝑜) given by Eq. 4.32 and
adding the stress terms induced by varying pore pressure caused by fluid flow (Eq. 4.21),
one can get the total stress distribution around wellbore with the presence of mudcake
100
2 4 2 2
,2 4 2 2
2 4 2
,2 4 2
2
z,2
21 1 3 4 cos 2 ln
2 2 3
21 1 3 cos 2 ln
2 2 3
2 cos 2
H h o H h o o o or i r p
i
H h o H h o o oi p
i
oz v H h
R R R R RYP
r r r R r
R R R RYP
r r R r
R
r
p
(4.33)
Total stress at the wellbore wall (mud-rock interface) 𝑟 = 𝑅𝑜 is
,
,
z,
2ln
3
22 cos 2 ln
3
2 cos 2
or i r p
i
oH h H h i p
i
z v H h p
RYP
R
RYP
R
(4.34)
The minimum total tangential stress occurs at 𝜃 = 0 𝑜𝑟 𝜋 on wellbore wall, i.e.
,min ,
23 ln
3
oh H i p
i
RYP
R
(4.35)
Rewrite this equation as
,min , ,3 h H i pl pP (4.36)
with
,
2ln
3
opl
i
RY
R
(4.37)
is the total tangential stress induced by plastic deformation of mudcake, and 𝜎𝜃,𝑝 is the
total tangential stress induced by fluid flow (varying pore pressure).
Regarding the typical wellbore radius 𝑅𝑜 in petroleum applications is between 0.1
and 0.13 m (Tran et al., 2011) and the typical mudcake thickness is between 0.002 and
0.006 m (Bezemer and Havenaar, 1966; Chenevert and Dewan, 2001; Griffith and
Osisanya, 1999; Sepehrnoori et al., 2005), i.e. 𝑅𝑖 is between 0.094 and 0.098 m, the value
of the term 𝑙𝑛(𝑅𝑜 𝑅𝑖⁄ ) is very small. Since the mudcake yield strength 𝑌 is also usually
101
very small (Cerasi et al., 2001; Cook et al., 2016), we can conclude from Eq. 4.37 that the
effect of mudcake strength on the wellbore tangential stress should be very small.
4.2.2.4 Effective Stress around Wellbore
Effective stress on wellbore wall can be determined by subtracting pore pressure
(Eq. 4.2) from the total stresses (Eq. 4.34) on wellbore wall and expressed as
'
,
'
,
'
z,
2ln
3
22 cos 2 ln
3
2 cos 2
or i r p o
i
oH h H h p o
i i
z v H h p o
RYP P
R
RYP P
R
P
(4.38)
The minimum effective tangential stress on wellbore wall can be found at 𝜃 =
0 𝑜𝑟 𝜋 as
'
,min ,
23 ln
3
oh H i p o
i
RYP P
R
(4.39)
or '
,min , ,3 h H i pl p oP P (4.40)
4.2.2.5 Fracture Pressure
Fracture pressure is determined based on a tensile failure criterion that fracture
occurs when the minimum effective stress on wellbore wall reaches the tensile strength of
the rock. Usually tensile strength of sedimentary rock is very small, so in this study we
simply ignore the tensile strength of the formation. Therefore, fracture occurs when
𝜎𝜃,𝑚𝑖𝑛′ = 0. 𝜎𝜃,𝑚𝑖𝑛
′ is given by Equation 4.39. Pore pressure 𝑃𝑜 at wellbore wall in Eq.
4.39 is given by Eq. 4.2. However, we still need to determine the fluid flow induced
tangential stress term 𝜎𝜃,𝑝 in Eq. 4.39. This term can be obtained from Eq. 4.21 by setting
𝑟 = 𝑅𝑜 as
102
2
2
, 1 22 2 2
22 op o o o o o
o e o
RR I R R P R I R
R R R
(4.41)
where, 𝐼1(𝑅0), 𝐼2(𝑅0), and ∆𝑃(𝑅0) can be determined from Eqs. 4.19, 4.20, and 4.12
respectively as
1 0oI R (4.42)
2 2
2 2 2 2
2
1 1 1 12ln 1 2ln 1
2 4 4 4 4
e o e eo e o e o o
o o
R R R RI R A R R AB R R P
R R
(4.43)
1 ln eo o
o
RP R AB P
R
(4.44)
Define
2 2' 2 2 2 21 1 1 1
2ln 1 2ln 12 4 4 4 4
e o e ee o e o
o o
R R R RM A R R AB R R
R R
(4.45)
1 ln e
o
RN AB
R
(4.46)
Then Eqs. 4.43 and 4.44 can be shortened as
'
2 o oI R M P (4.47)
o oP R N P (4.48)
Substituting Eqs. 4.42, 4.47 and 4.48 into Eq. 4.41, one can get the tangential stress
induced by fluid penetration at wellbore wall
'
, 2 2
22p o o
e o
MR N P
R R
(4.49)
Define '
2 2
2
e o
MM
R R
(4.50)
Eq. 4.49 is then rewritten as
, 2 2p o o i eR M N P M N P P (4.51)
Substituting Eqs. 4.2 and 4.51 into Eq. 4.39, the minimum effective tangential stress on the
wellbore wall can be determined as
103
'
,min
23 2 2 2 ln
3
oh H e i
i
RYM N B P M N B P
R (4.52)
Applying the tensile failure criterion, the fracture pressure of wellbore with mudcake is
determine as
23 2 ln
3
2 2
oh H e
if
RYM N B P
RP
M N B
(4.53)
This solution takes into account the effects of mudcake permeability, thickness and
strength on fracture pressure of a wellbore plastered by a layer of mudcake. In the following
section, sensitivity studies are performed to investigate the influences of these factors on
wellbore stress and fracture pressure, and the implications on lost circulation prevention
are discussed.
4.2.3 Example Cases
In this section, the effects of the mudcake thickness, permeability and strength on
wellbore stress and fracture pressure are analyzed through several example cases using the
analytical model presented in the above section.
Typical mudcake parameters used in this study are obtained from a literature
review. Final mudcake thickness with dynamic filtration in drilling may range between 2
to 6 mm (Bezemer and Havenaar, 1966; Chenevert and Dewan, 2001; Griffith and
Osisanya, 1999; Sepehrnoori et al., 2005). The permeability of mudcake may range
between 10−4 to 10−2 mD (Bezemer and Havenaar, 1966; Sepehrnoori et al., 2005).
The yield strength of mudcake may range between 0.1 to 0.6 MPa for oil based mud (Cook
et al., 2016). In the following case studies, mudcake parameters are manipulated within
these typical ranges for sensitivity analyses. The parameters are reported in Table 4.1, with
a set of base case and several supplementary cases for sensitivity study purposes.
104
Table 4.1: Summary of the parameters used in the example cases.
General Inputs Mudcake Parameters
Wellbore
Radius, m 0.1
Maximum Horizontal
Stress, MPa 22 Parameters
Base
Case
Supplementary
Cases
Outer
Radius, m 1
Minimum Horizontal
Stress, MPa 18
Mudcake
Thickness,
mm
2 4 6
Formation
Pressure,
MPa
10 Overburden Stress,
MPa 25
Mudcake
Permeability,
mD
0.01 0.001 0.0001
Wellbore
Pressure,
MPa
12 Angle θ, (o) 0 MudcakeYield
Strength, MPa 0.15 0.25 0.5
Rock
Permeability,
mD
1 Parameter η 0.3
4.2.3.1 Effect of Mudcake Thickness
Using the analytical solution derived in the above section, the pore pressure, fluid
flow induced total and effective tangential stresses, and fracture pressure with different
mudcake thickness are calculated. The results are shown in Figures 4.2 through 4.5. The
following conclusions about the effect of mudcake thickness are made based on the results:
Pore pressure around wellbore decreases with the increase of mudcake thickness as
shown in Figure 4.2. Even with a very thin (e.g. 2mm) mudcake, the pore pressure
at or close to the wellbore wall can have a significant decline compared with the
case without mudcake.
In the near wellbore region, total tangential stress induced by fluid flow is positive,
and decreases with the increase of mudcake thickness; adversely, away from the
wellbore, the induced stress is negative and increases with the increase of mudcake
thickness, as shown in Figure 4.3.
105
Figure 4.4 shows that the effective tangential stress induced by fluid flow
(compared with no flow case) is negative (tensile). Its value increases (tension
decreases) with mudcake thickness increase, meaning the thicker the mudcake, the
less likely for tensile fracture and thus lost circulation to occur.
With mudcake thickness increase, fracture pressure of the wellbore has significant
increase as shown in Figure 4.5, which means the wellbore is effectively
strengthened by mudcake. In this particular case, fracture pressure can be increased
by 2 MPa with a 4 mm mudcake on wellbore wall. This confirms the conclusion
from experimental studies that mudcake can effectively increase the pressure that a
wellbore can sustain and reduce the risk of lost circulation (Cook et al., 2016).
Figure 4.2: Pore Pressure distribution around wellbore with different mudcake thickness
𝑤.
106
Figure 4.3: Total tangential stress induced by fluid flow with different mudcake thickness
𝑤.
Figure 4.4: Effective tangential stress induced by fluid flow (compared with no flow case)
with different mudcake thickness 𝑤.
107
Figure 4.5 Fracture pressure with different mudcake thickness 𝑤.
4.2.3.2 Effect of Mudcake Permeability
Pore pressure around wellbore, fluid flow induced total and effective tangential
stresses, and fracture pressure with different mudcake permeability are calculated and
shown in Figures 4.6 through 4.9. The following conclusions about the effect of mudcake
permeability are made based on the results:
Pore pressure around wellbore decreases with the decrease of mudcake
permeability as shown in Figure 4.6. In this specific case, the pore pressure around
wellbore almost does not change (i.e. equal to formation pressure) with an
extremely low mudcake permeability of 10−4 mD. This case can practically be
approximated as the case of an impermeable wellbore.
In the near wellbore region, total tangential stress induced by fluid flow is positive,
and decreases with the decrease of mudcake permeability; but in the region away
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from wellbore, the induced stress is negative and increases with the decrease of
mudcake permeability, as shown in Figure 4.7.
Figure 4.8 shows the effective tangential stress induced by fluid flow (compared
with no flow case) is negative (tensile). Its value increases (tension decreases) with
the decrease of mudcake permeability, meaning the lower the permeability of the
mudcake, the less likely for tensile fracture and thus lost circulation to occur. In
this specific case, there is almost no tensile tangential stress induced by fluid flow
with a mudcake permeability as low as 10−4 mD.
With mudcake permeability decrease, fracture pressure of the wellbore has
significant increase as shown in Figure 4.9, which means the wellbore can be
effectively strengthened by a low-permeability mudcake. In this particular case,
fracture pressure can be increased by 2 MPa by changing the mudcake permeability
from 10−2 to 10−3 mD. Mudcake permeability depends on mud type, additives
or LCMs in the mud, and bottom hole conditions. It is critical to properly engineer
the mud formulations to build a low-permeability mudcake on wellbore wall in
preventive lost circulation treatments.
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Figure 4.6: Pore Pressure distribution around wellbore with different mudcake
permeability 𝐾1.
Figure 4.7: Total tangential stress induced by fluid flow with different mudcake
permeability 𝐾1.
110
Figure 4.8: Effective tangential stress induced by fluid flow (compared with no flow case)
with different mudcake permeability 𝐾1.
Figure 4.9: Fracture pressure with different mudcake permeability 𝐾1.
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4.2.3.3 Effect of Mudcake Strength
Figures 4.10 and 4.11 show the effective tangential stress and fracture pressure of
wellbore with different mudcake (yield) strength. It can be seen from Figure 4.10 that with
the typical values of mudcake yield strength (less than 1 MPa), the effect of mudcake
strength on wellbore stress is extremely small. Figure 4.11 shows that with the increase of
mudcake strength, fracture pressure has a very slight increase, but this increase is negligibly
small for practical considerations. So it can be concluded that the strength of mudcake itself
almost does not contribute to the “strength” of wellbore. Mudcake “strengthens” the
wellbore mainly through preventing/mitigating fluid seepage from wellbore to the
surrounding formation and therefore inhibiting the development of tensile stress around
wellbore.
Figure 4.10: Effective tangential stress around wellbore with different mudcake yield
strength 𝑌.
112
Figure 4.11: Fracture pressure with different mudcake yield strength 𝑌.
4.3 A NUMERICAL MODEL FOR TIME-DEPENDENT MUDCAKE
4.3.1 Objective and Challenge
The analytical model proposed in the above section assumes steady-state fluid flow,
so it does not take into account the time-dependent effect on preventive wellbore
strengthening treatments based on plastering the wellbore with mudcake. However, in
realistic situation, mudcake buildup is a time-dependent process. Three time-dependent
processes should be considered in modeling the dynamic mudcake problem:
Time-dependent fluid flow in the porous rock and mudcake
Time-dependent mudcake thickness buildup
Time-dependent mudcake permeability reduction
Therefore, in this section, in order to take into account these time-dependent effects,
a numerical model is developed to simulate the evolutions of near-wellbore stress and pore
pressure with time-dependent fluid flow and mudcake parameters based on finite-element
method using Abaqus® software and FORTRAN® subroutines.
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The challenge in developing such a finite-element model is that it is difficult to
actually manipulate the mudcake thickness to model mudcake buildup during the
simulation, because with the current finite-element techniques it is difficult to dynamically
and smoothly change the size of the elements or add new elements that represent the
mudcake during the simulation process. However, it is more practical to manipulate the
material parameters (e.g. permeability) as a function of time during the simulation.
Therefore, in this work, an equivalent method is used to describe the effects of both time-
dependent mudcake thickness and dynamic mudcake permeability with one material
parameter – “equivalent mudcake permeability”. By manipulating this single parameter
during the simulation, we can get the same results as simultaneously changing both
mudcake thickness and permeability, but in a more practical and easier way. In the
following subsections, the equivalent method and the finite-element model development
are described in detail.
4.3.2 An Equivalent Method
Since it’s very difficult to dynamically change the mudcake element size, thus the
mudcake thickness during the simulation process of a finite-element model, a constant
mudcake thickness will be defined in our numerical model. However, an equivalent method
will be used to capture the effects of both mudcake thickness buildup and permeability
change with time using a single parameter – equivalent mudcake permeability.
We still consider the mudcake, wellbore and formation are in a plane-strain
condition as shown in Figure 4.12. In real situation while drilling, as mentioned above,
both mudcake thickness 𝑤(𝑡) and permeability 𝑘(𝑡) are functions of time. But in the
numerical model, we assume the mudcake has a constant thickness 𝑤𝑜 and an equivalent
114
permeability 𝑘𝑒(𝑡). The equivalent permeability 𝑘𝑒(𝑡) is a function of both real mudcake
thickness 𝑤(𝑡) and permeability 𝑘(𝑡).
Figure 4.12: Illustration of mudcake model with constant mudcake thickness 𝑤𝑜 and
equivalent mudcake permeability 𝑘𝑒(𝑡).
The equivalent mudcake permeability 𝑘𝑒(𝑡) is determined by achieving the same
fluid flow rate under the same differential pressure across the mudcake between the
numerical model with constant mudcake thickness and the real case with dynamic mudcake
thickness. Therefore, using Darcy’s law for both cases we can have 2𝜋𝐾𝑒(𝑡)
𝜇𝑙𝑛𝑅𝑖+𝑤𝑜
𝑅𝑖
∆𝑝 =2𝜋𝐾(𝑡)
𝜇𝑙𝑛𝑅𝑖+𝑤(𝑡)
𝑅𝑖
∆𝑝 (4.54)
The left side of Eq. 4.54 is the term for the model with constant mudcake thickness 𝑤𝑜
and equivalent mudcake permeability 𝑘𝑒(𝑡); while the right side of Eq. 4.54 is the term
for the real situation with time-dependent mudcake thickness 𝑤𝑜 and permeability 𝑘𝑒(𝑡).
The equivalent mudcake permeability 𝑘𝑒(𝑡) is then determined as
𝐾𝑒(𝑡) = 𝐾(𝑡)𝑙𝑛(1+
𝑤𝑜𝑅𝑖
)
𝑙𝑛(1+𝑤(𝑡)
𝑅𝑖) (4.55)
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Once the real time-dependent mudcake thickness 𝑤(𝑡) and permeability 𝑘(𝑡)
are known (e.g. from experimental study), a time-dependent equivalent permeability 𝑘𝑒(𝑡)
can be determined using Eq. 4.55 and then assigned as a material parameter to the mudcake
in the finite-element model. It can be seen from Eq. 4.55 that the equivalent mudcake
permeability reflects not only the time-dependent mudcake thickness but also the time-
dependent mudcake permeability.
4.3.3 Model Formulation
4.3.3.1 Model Geometry
Mudcake on a vertical wellbore is considered. The problem is still assumed to be
in 2D plane-strain condition. Due to symmetry, only one quarter of the wellbore is modeled
as shown in Figure 4.13. The wellbore radius is 0.1m. Both of the length and the width of
the quarter model are 2m. Mudcake has a constant thickness of 3mm and is represented
with two layers of elements as shown in Figure 4.13. The mesh is refined in near wellbore
region. Both of the mudcake and formation are molded with coupled pore pressure and
deformation elements in Abaqus®.
Figure 4.13: Geometry of the finite-element mudcake model.
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4.3.3.2 Initial and Boundary Conditions
Symmetric boundary conditions are applied to the left and bottom boundaries of
the model as shown in Figure 4.14. The maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) and minimum
horizontal stress (𝑆ℎ𝑚𝑖𝑛) are applied to the right and top outer boundaries of the model,
respectively. Constant pore pressure boundary condition 𝑃𝑝𝑜 with its value equal to the
formation pore pressure is also applied to the outer boundaries. Bottom hole pressure 𝑃𝑤
is applied to inner surface of mudcake. Also defined on this surface is a pore pressure
boundary 𝑃𝑝𝑖 with its value equal to the bottom hole pressure 𝑃𝑤. Initial pore pressure
𝑃𝑝𝑜 is applied to the whole domain of the model. A set of boundary condition values is
given in Table 4.2 for the numerical simulation studies in the following sections.
Figure 4.14: Boundary conditions of the finite-element mudcake model.
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Table 4.2: Input boundary-condition values for the mudcake model.
Parameters Values
𝑆𝐻𝑚𝑎𝑥 22 MPa
𝑆ℎ𝑚𝑖𝑛 18 MPa
𝑃𝑝𝑜 10 MPa
𝑃𝑤 12 MPa
𝑃𝑝𝑖 12 MPa
4.3.3.3 Material Properties
Formation. The formation rock is modeled as poroelastic material with Young’s
modulus 𝐸𝑓 = 1.583 × 106𝑘𝑃𝑎 , Poisson’s ratio 𝑣𝑓 = 0.22 , porosity 𝜙𝑓 = 0.14 , and
permeability 𝑘𝑓 = 0.1 𝑚𝐷.
Mudcake. Mudcake is also modeled as poroelastic material. However, the
mudcake is considered to be very soft/flexible compared with the formation rock. For a
very soft mudcake, as shown in the analytical study in Section 4.2 the mechanical
properties or the “strength” of the mudcake has negligibly small effect on the stress and
pore pressure around wellbore. Therefore, proper mechanical properties should be selected
for modeling mudcake to make sure it alters near-wellbore stress only through its porous
properties (permeability and porosity), rather than through its mechanical properties. This
can be achieved by selecting a low Young’s modulus and a very high Poisson’s ratio for
the mudcake in the finite-element model presented in this section. This statement will be
validated later in Section 4.3.4. In this study, Young’s modulus 𝐸𝑚 = 1.583 × 105𝑘𝑃𝑎
and Poisson’s ratio 𝑣𝑚 = 0.49 is selected for mudcake and can satisfy the requirements.
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Experimental studies show that, with mud circulation in the wellbore, mudcake
thickness increases and finally approaches a maximum equilibrium value, while mudcake
permeability decreases and finally approaches a minimum equilibrium value (Bezemer and
Havenaar, 1966; Osisanya and Griffith, 1997; Tran et al., 2011). For this study, a typical
profile of mudcake thickness buildup with time given by Eq. 4.56 is assumed (Tran et al.,
2011). As shown in Figure 4.15, the mudcake approaches its maximum equilibrium
thickness of 3mm after about 4 hours’ exposure to circulating mud.
𝑤(𝑡) = 0.003(1 − 𝑒−0.0003𝑡) (4.56)
A typical profile of mudcake permeability reduction with time given by Eq. 4.57
(Tran et al., 2011) is used in this study. The mudcake has an initial permeability of 0.002
mD, which decreases to an equilibrium value of 0.001 mD after approximate 4 hours’ mud
circulation as shown in Figure 4.16.
𝑘(𝑡) = 0.001(1 + 𝑒−0.0003𝑡) (4.57)
Therefore, according to Eqs. 4.55, 4.56 and 4.57, the equivalent mudcake
permeability of the mudcake is
𝑘𝑒(𝑡) =0.001(1+𝑒−0.0003𝑡)𝑙𝑛(1+
𝑤𝑜𝑅𝑖
)
𝑙𝑛[1+0.003(1−𝑒−0.0003𝑡)
𝑅𝑖]
(4.58)
Figure 4.17 shows the variation of equivalent mudcake permeability with time. At
the early time, the equivalent permeability is very high because of the small mudcake
thickness (see Figure 4.15) and high mudcake permeability (see Figure 4.16) in real case.
Finally the equivalent permeability approaches 0.001 mD which is equal to the real
equilibrium permeability because in this example the real equilibrium mudcake thickness
is equal to the constant mudcake thickness defined in the model. The variation of equivalent
permeability with time given by Eq. 4.58 is coded into a FORTRAN® subroutine and
119
plugged into the Abaqus® model to simulate the dynamic evolution of mudcake thickness
and permeability.
Figure 4.15: Variation of mudcake thickness with time.
Figure 4.16: Variation of mudcake permeability with time.
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Figure 4.17: Variation of equivalent mudcake permeability with time.
4.3.3.4 Simulation Procedures
Three steps are defined in the model for the simulating the dynamic mudcake
problem.
Step 1. Geostatic step in Abaqus® is used to obtain initial stress and pore
pressure of the entire simulation domain. In this step, no wellbore is drilled and
the formation in unperturbed condition.
Step 2. A set of elements representing the wellbore is removed and mud
pressure is applied to the wellbore surface to simulate the drilling process and
obtain the stress and pore pressure distribution immediately after drilling.
Step 3. Mudcake with the equivalent permeability defined by Eq. 4.58 is added
onto the wellbore surface and the mud pressure is reassigned onto the mudcake
surface, and then fluid seepage is simulated up to 8 hours.
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4.3.4 Model Calibration
As mentioned in section 4.2, mudcake alters the stress and pore pressure around
wellbore mainly through preventing/mitigating fluid flow through wellbore to the
surrounding formation due to its low permeability. Because mudcake is extremely
soft/flexible compared with formation rock, the mechanical properties or the “strength” of
mudcake should not alter the stress and pore pressure around wellbore very much.
Therefore, when the mudcake is extremely permeable (with very high permeability), the
mud pressure in the wellbore can be totally transmitted onto the wellbore surface, and the
stress state around wellbore should be the same as that of a permeable wellbore without
mudcake. On the other hand, when the mudcake is perfectly impermeable (with zero
permeability), the stress state around wellbore should be the same as that of an
impermeable wellbore wall without mudcake. To calibrate the model, the pore pressure,
tangential stress and radial stress along 𝑆𝐻𝑚𝑎𝑥 direction are compared for the following
two sets of equivalent cases:
Set 1: Mudcake with extremely high permeability ( 107 Darcy) versus
permeable wellbore without mudcake.
Set 2: Impermeable mudcake (zero permeability) versus impermeable wellbore
without mudcake.
Figures 4.18 through 4.23 show the excellent agreements in pore pressure,
tangential stress and radial stress for the two sets of equivalent cases. The results validate
the solution behaviors of the proposed numerical model, and proved the statement that the
mechanical properties of a very soft mudcake with small modulus and high Poisson’s ratio
have negligibly small effect on wellbore stress.
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Figure 4.18: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for wellbore with
extremely permeable mudcake and permeable wellbore without mudcake.
Figure 4.19: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for
wellbore with extremely permeable mudcake and permeable wellbore without
mudcake.
123
Figure 4.20: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for
wellbore with extremely permeable mudcake and permeable wellbore without
mudcake.
Figure 4.21: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for wellbore with
impermeable mudcake and impermeable wellbore without mudcake.
124
Figure 4.22: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for
wellbore with impermeable mudcake and impermeable wellbore without
mudcake.
Figure 4.23: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 3 hours for
wellbore with impermeable mudcake and impermeable wellbore without
mudcake.
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4.3.5 Simulation Results: Time-Dependent Mudcake Effects
4.3.5.1 Time-Dependent Mudcake Thickness Buildup
In this section, the time-dependent effects of mudcake thickness buildup on pore
pressure and effective stresses around wellbore are analyzed using the proposed numerical
model. The mudcake permeability is assumed to be constant at 0.01 mD during the entire
filtration process. Three simplified conditions are also simulated and used as baselines for
comparison, including: (1) no mudcake on wellbore wall, (2) a perfect impermeable
mudcake on wellbore wall, and (3) a permeable mudcake with constant thickness of 3 mm
(equal to the maximum equilibrium thickness of the dynamic mudcake). Pore pressure,
effective radial stress and effective tangential stress along the direction of 𝑆𝐻𝑚𝑎𝑥 after
seepage periods of 30 minutes and 4 hours are shown in Figures 4.24 through 4.29.
Generally, the following effects of time-dependent mudcake thickness buildup are
observed from the simulation results:
At early time (t = 30 minutes), as shown in Figures 4.24, 4.26 and 4.28, the pore
pressure, effective radial stress and effective tangential stress (compression is
negative in this numerical study) with time-dependent mudcake thickness are larger
than those with constant mudcake thickness and closer to those without mudcake,
because the thickness is still small at the early time.
At late time (t= 4 hours), as shown in Figures 4.25, 4.27 and 4.29, as the mudcake
grows and approaches the equilibrium thickness of 3 mm, the pore pressure and
stress profiles approach those of the constant mudcake thickness case. Tangential
stress becomes more compressive compared with that at the early time t=30
minutes, which means mudcake buildup results in positive effect for lost circulation
prevention.
126
Figure 4.24: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
Figure 4.25: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
127
Figure 4.26: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
Figure 4.27: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
128
Figure 4.28: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
Figure 4.29: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
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4.3.5.2 Time-Dependent Mudcake Permeability Reduction
In this section, the time-dependent effects of mudcake permeability reduction on
pore pressure and stresses abound wellbore are analyzed using the proposed numerical
model. The mudcake thickness is assumed to be constant at 3 mm during the entire filtration
process. Three simplified conditions are also simulated and used as baselines for
comparison, including: (1) no mudcake on the wellbore wall, (2) a perfect impermeable
mudcake on wellbore wall, and (3) permeable mudcake with constant permeability of 0.001
mD which is equal to the minimum equilibrium permeability of the dynamic mudcake.
Pore pressure, effective radial stress and effective tangential stress along the direction of
𝑆𝐻𝑚𝑎𝑥 after seepage periods of 30 minutes and 4 hours are shown in Figures 4.30 through
4.35.
Generally, the following effects of time-dependent mudcake permeability reduction
are observed from the simulation results:
At early time (t = 30 minutes), as shown in Figures 4.30, 4.32 and 4.34, the pore
pressure, effective radial stress and effective tangential stress (compression is
negative) with time-dependent mudcake thickness are larger than those with
constant mudcake thickness and closer to those without mudcake, because the
permeability is still large at early time.
As the mudcake permeability decreases and approaches the equilibrium
permeability of 0.01 mD at t = 4 hours, as shown in Figures 4.31, 4.33 and 4.35,
the pore pressure and stress profiles approach those of the constant (minimum)
mudcake permeability case. Tangential stress becomes more compressive
compared with that at the early time t=30 minutes, which means mudcake
permeability reduction with time results in positive effect for lost circulation
prevention.
130
Figure 4.30: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
Figure 4.31: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
131
Figure 4.32: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
Figure 4.33: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
132
Figure 4.34: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
Figure 4.35: Effective tangential stress profiles along SH direction at t = 4 hours.
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4.3.5.3 Coupled Time-Dependent Mudcake Thickness Buildup and Permeability
Reduction
In this section, the coupled effects of time-dependent mudcake thickness buildup
and permeability reduction on pore pressure and stresses around wellbore are analyzed
using the proposed numerical model. Three simplified conditions are also simulated and
used as baselines for comparison, including: (1) no mudcake on the wellbore wall, (2) a
perfect impermeable mudcake on wellbore wall, and (3) a permeable mudcake with
constant permeability of 0.001 mD (equal to the minimum equilibrium permeability of the
dynamic mudcake) and constant thickness of 3 mm (equal to the maximum equilibrium
thickness of the dynamic mudcake) on wellbore wall. The results are shown in Figures 4.36
through 4.41.
Generally, the following effects of coupled time-dependent mudcake thickness
buildup and permeability reduction are observed from the simulation results:
At early time (t = 30 minutes), as shown in Figures 4.36, 4.38 and 4.40, the pore
pressure, effective radial stress and effective tangential stress (compression is
negative) with time-dependent mudcake thickness and permeability are larger than
those with constant mudcake thickness and permeability and closer to those without
mudcake, because the mudcake thickness is still small and the permeability is still
large at this time. At early time, assuming mudcake with no permeability
(equivalent to impermeable wellbore) or with constant final thickness and
permeability may lead to substantial underestimation of pore pressure and effective
stress around wellbore wall.
At late time (t = 4 hours), as shown in Figures 4.37, 4.39 and 4.41, as the mudcake
approaches the equilibrium thickness of 3 mm and equilibrium permeability of
0.001mD, the pore pressure and stress profiles approach those with constant
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mudcake thickness and permeability. Tangential stress becomes more compressive
compared with that at the early time, which means the coupled effect of mudcake
thickness buildup and permeability reduction with time results in positive effect for
lost circulation prevention.
Figure 4.36: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
135
Figure 4.37: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
Figure 4.38: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
136
Figure 4.39: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 4 hours.
Figure 4.40: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
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4.3.5.4 Significance of Mudcake Permeability
In this section, the effects of mudcake permeability on near-wellbore pore pressure
and stress distributions are examined. The results with mudcake permeability of 0.01 mD
and 0.001 mD are compared (mudcake thickness is constant at 3 mm). Two simplified
conditions are also simulated and used as baselines for comparison, including: (1) no
mudcake on the wellbore wall, and (2) a perfect impermeable mudcake on wellbore wall.
The comparison results are shown in Figures 4.42 through 4.44.
Generally, the following effects of mudcake permeability on near-wellbore pore
pressure and stresses are observed from the simulation results:
As expected, the pore pressure with smaller mudcake permeability of 0.001 mD
are much smaller than those with larger mudcake permeability of 0.01 mD and
much closer to those with impermeable mudcake.
The compressive tangential stress with smaller mudcake permeability of 0.001
mD are much larger than those with larger mudcake permeability of 0.01 mD
and much closer to those with impermeable mudcake, which implies that
fractures or lost circulation events are more unlikely to occur with low-
permeability mudcake due to the larger compressive tangential stress.
Therefore, it is important to build a low-permeability mudcake on wellbore wall
as soon as possible when taking preventive wellbore strengthening treatments
during drilling.
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Figure 4.42: Pore pressure profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
Figure 4.43: Effective radial stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
140
Figure 4.44: Effective tangential stress profiles along 𝑆𝐻𝑚𝑎𝑥 direction at t = 30 minutes.
4.4 SUMMARY
In this chapter, an analytical solution for the effects of mudcake thickness,
permeability and strength on the wellbore stress and fracture pressure in a steady flow
condition is presented. Sensitivity studies are then performed using this solution. The
results show that both mudcake thickness and permeability have great influence on
wellbore stress and fracture pressure, while the effect of mudcake strength is negligibly
small for practical consideration. In particular, an increasing of fracture pressure with
decreasing mudcake permeability and/or increasing mudcake thickness is observed.
For a further step, a finite-element model taking into account the transient effects
of mudcake thickness buildup coupled with mudcake permeability reduction on near-
wellbore stress state is developed. A one-parameter description of both mudcake thickness
buildup and permeability reduction based on an equivalent flow study is used to facilitate
the model development without compromising the calculation accuracy. The results show
that taking into account the time-dependent mudcake thickness buildup and permeability
141
reduction results in a wellbore stress state between that without considering mudcake effect
and that assuming an impermeable mudcake (or that assuming constant final-equilibrium
mudcake thickness and permeability). The numerical model developed in this chapter
presents a useful tool to analyze the time-dependent stress evolution around wellbore with
dynamic mudcake development for the design and evaluation of preventive wellbore
strengthening treatments based on plastering wellbore surface with mudcake.
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CHAPTER 5: Modeling Study of Remedial Wellbore Strengthening
Treatments2
Remedial wellbore strengthening methods attempt to “strengthen” the wellbore by
bridging/plugging lost circulation fractures with LCMs after fluid loss already occurs. It is
an effective method to mitigate or stop fluid loss in the drilling industry. Although a
number of successful applications have been reported, remedial wellbore strengthening
operations are still mostly performed on a trial-and-error basis, without a clear
understanding of its fundamentals. The effects of several parameters are still not thoroughly
understood. Thus, for better understanding of the underlying mechanisms of remedial
wellbore strengthening treatments based on bridging lost circulation fractures, this chapter
performs studies on it based on both analytical and numerical modeling. The analytical
model derived based on linear elastic fracture mechanics provides a fast procedure to
predict fracture pressure change before and after bridging the fractures, while the numerical
model developed using finite-element method gives a more detailed description of the
distribution of local stress and fracture width with remedial wellbore strengthening
operations.
2 Parts of this chapter have been published in the following journal papers which were supervised by K. E.
Gray:
Feng, Y., Arlanoglu, C., Podnos, E., Becker, E., Gray, K.E., 2015. Finite-Element Studies of Hoop-
Stress Enhancement for Wellbore Strengthening. SPE Drill. Complet. 30, 38–51. doi:10.2118/168001-
PA.
Feng, Y., Gray, K.E., 2016a. A parametric study for wellbore strengthening. J. Nat. Gas Sci. Eng. 30,
350–363. doi:10.1016/j.jngse.2016.02.045.
Feng, Y., Gray, K.E., 2016b. A fracture-mechanics-based model for wellbore strengthening applications.
J. Nat. Gas Sci. Eng. 29, 392–400. doi:10.1016/j.jngse.2016.01.028.
143
5.1 INTRODUCTION
Remedial wellbore strengthening methods attempt to “strengthen” the wellbore
after fractures has already occurred on wellbore. It is an effective method to mitigate or
stop fluid loss by bridging lost circulation fractures with LCMs. The ultimate objective of
remedial wellbore strengthening treatments is to increase the fracture pressure that a
wellbore can sustain without significant fluid loss and widen the drilling mud weight
window. Various experimental studies on remedial wellbore strengthening have been
carried out in the drilling industry.
In laboratory and field experimental studies, repeated leak-off tests are commonly
used to investigate the effectiveness of remedial wellbore strengthening treatments. By
performing repeated leak-off tests in a wellbore with and without LCMs in the injection
fluid, the fracture pressure for strengthened and un-strengthened wellbore can be
compared. A number of laboratory experiments (Aston et al., 2007, 2004; Guo et al., 2014;
Morita et al., 1996a; Onyia, 1994; Savari et al., 2014) and field applications (Alberty and
McLean, 2004; Dupriest, 2005; van Oort et al., 2011) have shown that fracture pressure
can be effectively increased by bridging small fractures in remedial wellbore strengthening
operations. Figure 5.1 compares the fracture pressure of a wellbore without and with
wellbore strengthening treatment in one experimental study of the DEA 13 project (Onyia,
1994). The wellbore was first fractured by a leak-off test with drilling fluid free of LCMs,
and then a repeated test was conducted with drilling fluid containing LCMs. It is shown
that the wellbore was effectively strengthened with an increase of fracture (breakdown)
pressure about 5000 psi after applying remedial wellbore strengthening treatments.
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Figure 5.1: Comparison of fracture pressure with and without remedial wellbore
strengthening treatment in DEA 13 experimental study (reproduced from
Onyia, 1994).
Figure 5.2 is the result of an experimental study on remedial wellbore strengthening
performed by Guo et al. (2014). In this test drilling fluid without LCMs was first injected
to fracture the intact wellbore and a fracture (breakdown) pressure of 870 psi was achieved.
Subsequently, two repeated leak-off tests using the same drilling mud without LCMs were
conducted to test the strength of the wellbore with existing fractures created in the first test.
The fracture (breakdown) pressure in these two case was about 400 psi lower than in the
intact wellbore. Subsequently, in the final test the drilling mud was replaced with base
drilling mud plus 30-lb/bbl graphitic LCMs to strengthen the wellbore. It is observed that
an enhanced fracture (breakdown) pressure about 1700 psi was achieved in this case, which
was about 800 and 1200 psi higher than the fracture (breakdown) pressure in the intact
wellbore and fractured wellbore, respectively.
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Figure 5.2: Results of a laboratory wellbore strengthening test (reproduced from Guo et al.,
2014).
Figure 5.3 shows a field wellbore strengthening test by Aston et al. (2004). A base
mud free of LCM was first pumped to fracture a vertical wellbore, and an original
breakdown pressure of 1200 psi was observed. In a following test, the base mud was
replaced by a designed mud containing 80-lb/bbl LCM solids to investigate the effect of
remedial wellbore strengthening. The solid curve in Figure 5.3 shows the pressure-time
curve using LCMs. A fracture (breakdown) pressure about 2050 psi was reached in this
case, which is about 850 psi higher than the original state. This significant increase in
fracture (breakdown) pressure clearly indicates that the fractures can be successfully
bridged using remedial wellbore strengthening treatment.
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Figure 5.3: Results of LOT field tests before and after taking remedial wellbore
strengthening treatment (reproduced from Aston et al., 2004).
However, despite successful lab experiments and field applications, the
fundamental physics of remedial wellbore strengthening are not thoroughly understood. A
lot of disagreements still exist in the drilling industry. There is still a lack of proper
mathematic models to quantitatively describe this problem. Thus, for better understanding
of the underlying mechanisms of remedial wellbore strengthening treatments based on
bridging lost circulation fractures, studies using both analytical and numerical modeling
have been carried out in this chapter. The analytical model proposed based on linear elastic
fracture mechanics provides a fast procedure to predict fracture pressure change before and
after bridging the fractures, while the numerical model developed using finite-element
method gives a more detailed description of the distribution of local stress and fracture
width with remedial wellbore strengthening operations.
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5.2 A FRACTURE-MECHANICS-BASED ANALYTICAL MODEL FOR REMEDIAL
WELLBORE STRENGTHENING APPLICATIONS
Based on linear elastic fracture mechanics, this section illustrates an analytical
solution for investigating geomechanical aspects of remedial wellbore strengthening
operations. The proposed solution (1) provides a fast procedure to predict fracture pressure
before and after bridging a preexisting fracture in remedial wellbore strengthening
treatments, and (2) considers the effects of wellbore-fracture geometry, in-situ stress
anisotropy, and LCM bridge location. The solution is validated by comparison with
available fracture mechanics examples, then used to investigate and quantify the effect of
several parameters on wellbore strengthening with a sensitivity study. Results show that
the wellbore can be effectively strengthened by increasing fracture pressure with remedial
wellbore strengthening operations, and the magnitude of strengthening is affected by LCM
bridge location, in-situ stress anisotropy, and formation pore pressure. The proposed
solution illustrates how remedial wellbore strengthening treatment works, and provides
useful considerations for field operations.
In addition, in order to avoid lost circulation, it is crucial to adjust mud weight
during the drilling in a timely manner, according to fracture pressure. Some essential
parameters controlling fracture pressure, such as geometry of pre-existing fractures on the
wellbore, are usually not known prior to drilling, but they may be measured while drilling.
Therefore, performing real-time analyses is important in order to make prompt adjustments
to drilling operations. The advantages of an analytical model compared to a numerical
model for real-time analyses are obvious, because it is usually very concise and easy to
implement, and building a meshed structure for conducting a large number of iterative
computations is not needed, contrary to numerical analyses, such as finite-element and
boundary-element analyses. The proposed analytical model provides a rapid procedure to
148
predict fracture pressure before and after bridging a preexisting fracture. It has the potential
for implementation in near-real time drilling analysis for wellbore strengthening evaluation
and mud weight adjustments.
5.2.1 Analytical Model
5.2.1.1 Model Description
Figure 5.4 illustrates the problem under consideration. It is a plane-strain wellbore
model with two short fractures extending symmetrically from the wellbore wall and
perpendicular to the minimum horizontal stress. Before bridging the fracture (i.e. without
the bridges in Figure 5.4), given the small length of the fractures, the pressure inside the
fracture is assumed to be uniform and equal to the wellbore pressure or the equivalent
circulation pressure during drilling. After bridging the fracture at some location inside the
fracture shown in Figure 5.4, the pressure ahead of the bridge from the fracture mouth to
the bridge is equal to wellbore pressure. Pressure behind the bridge is equal to formation
pore pressure, assuming the bridge perfectly blocks pressure transmission in the fracture;
pressure higher than formation pore pressure behind the bridge will bleed off to pore
pressure due to fluid leak-off into the porous rock. In the following discussion, the fracture
portions ahead of and behind the bridge are referred to as invaded zone and non-invaded
zone, respectively.
In this analytical study, a goal is to develop a fast and easy-to-use analytical solution
to quantify fracture pressure change after bridging the fracture. The solution is derived by
combining Kirsch stress solutions around a circular borehole with a linear-elastic fracture
criterion. The following sections detail the procedures for deriving the solution.
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Figure 5.4: Schematic of remedial wellbore strengthening problem.
5.2.1.2 Net Pressure along Fracture
The system shown in Figure 5.4 can be decomposed into two simper cases
(Carbonell and Detournay, 1995; Shahri et al., 2014) as shown in Figure 5.5: (1) an intact
wellbore subject to wellbore pressure and anisotropic horizontal stresses, and (2) a wellbore
with two symmetric fractures subject solely to pressure loads inside the fracture. The
tangential stress along the fracture direction in the first case can be determined by Kirsch
solution:
𝑆𝜃𝜃 =1
2(𝑆𝐻𝑚𝑎𝑥 + 𝑆ℎ𝑚𝑖𝑛) (1 +
𝑅2
𝑟2) −
1
2(𝑆𝐻𝑚𝑎𝑥 − 𝑆ℎ𝑚𝑖𝑛) (1 + 3
𝑅4
𝑟4) − 𝑃𝑤
𝑅2
𝑟2 (5.1)
where, 𝑆𝜃𝜃 is the tangential stress along the fracture direction; 𝑆𝐻𝑚𝑎𝑥 and 𝑆ℎ𝑚𝑖𝑛 are the
maximum and minimum horizontal stresses, respectively; 𝑅 is the wellbore radius; 𝑟 is
the radial distance from the wellbore center; 𝑃𝑤 is the wellbore pressure.
150
Figure 5.5: Problem decomposition. Left: an intact wellbore subject to wellbore pressure
and anisotropic far-field horizontal stresses; Right: wellbore with two
symmetric fractures subject to fluid pressure on fracture surfaces.
The net pressure acting on the fracture surfaces is defined as the difference between
the fluid pressure inside the fracture and the tangential stress in the rock normal to the
fracture surfaces. Assuming short fracture length, the tangential stress solution along the
fracture direction given by Eq. 5.1 is still valid. The net pressure of the fracture can be
approximated as:
𝑃𝑛𝑒𝑡 = 𝑃𝑓𝑙𝑢𝑖𝑑 − 𝑆𝜃𝜃 (5.2)
where 𝑃𝑛𝑒𝑡 is the net pressure along the fracture; 𝑃𝑓𝑙𝑢𝑖𝑑 is the fluid pressure inside the
fracture; it is equal to wellbore pressure and formation pore pressure in the invaded zone
and non-invaded zone, respectively. Therefore, the net pressure in the invaded zone after
bridging the fracture is:
𝑃𝑤𝑒𝑡 = 𝑃𝑤 − 𝑆𝜃𝜃 (5.3)
where, 𝑃𝑤𝑒𝑡 is the net pressure in the invaded zone from fracture mouth to bridge location.
The net pressure in the non-invaded zone is:
𝑃𝑑𝑟𝑦 = 𝑃𝑝 − 𝑆𝜃𝜃 (5.4)
151
where 𝑃𝑑𝑟𝑦 is the net pressure in the non-invaded zone from bridge location to fracture
tip; 𝑃𝑝 is formation pore pressure.
5.2.1.3 Solution for Fracture with Point Load
Figure 5.6 shows a fracture in an elastic solid, subject to a couple of point loads
normal to the fracture surfaces. The fracture length is 𝑎, and the distance from the loading
positon to the fracture mouth is 𝑏. Based on linear-elastic fracture mechanics theory, an
approximate solution of fracture-tip stress intensity factor for this problem is (Tada et al.,
1985):
𝐾𝐼0 =2𝑃
√𝜋𝑎∙ 𝐹 (
𝑏
𝑎) (5.5)
where, 𝐾𝐼0 is the fracture-tip stress intensity factor of the fracture subject to a couple of
point loads; 𝑃 is the magnitude of the points loads; 𝐹 (𝑏
𝑎) =
1.3−0.3(𝑏
𝑎)
5 4⁄
√1−(𝑏
𝑎)
2 , is a function
of fracture geometry and loading positon; 𝑎 is the fracture length; and 𝑏 is the distance
from the loading position to the fracture mouth.
Figure 5.6: Fracture in an elastic solid subject to a couple of point loads.
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5.2.1.4 Solution for Fracture with Distributed Load
In this study, it is assumed that the length of the pre-existing fracture emanating
from the wellbore is relatively small compared to the wellbore radius. The fracture-tip
stress intensity factor solution of the wellbore strengthening problem depicted in Figure
5.4 can be found by integrating the point-load solution in Eq. 5.5 from the fracture mouth
to the fracture tip, with the point load replaced by the net pressure at each location. Thus,
the fracture-tip stress intensity factor for the wellbore strengthening problem in Figure 5.4,
with distributed net pressure on fracture surfaces, can be obtained by calculating the
following integrals:
𝐾𝐼 = ∫2𝑃𝑤𝑒𝑡
√𝜋𝑎
𝐷
𝑅𝐹 (
𝑏
𝑎) 𝑑𝑟 + ∫
2𝑃𝑑𝑟𝑦
√𝜋𝑎
𝐿
𝐷𝐹 (
𝑏
𝑎) 𝑑𝑟 (5.6)
𝑎 = 𝐿 − 𝑅 (5.7)
𝑏 = 𝑟 − 𝑅 (5.8)
where 𝐾𝐼 is the fracture-tip stress intensity factor for the fracture with distributed load; 𝐷
is the radial distance from the bridge location to the wellbore center; 𝐿 is the radial
distance from the fracture tip to wellbore center; 𝑟 is the radial distance from a certain
point along the fracture to the wellbore center.
By substituting Eqs. 5.3 and 5.4 into Eq. 6 and computing the integrals, the
following solution for the fracture-tip stress intensity factor can be obtained:
𝐾𝐼 = (𝐹1 + 𝐹2) ∙ [2𝑃𝑤 − (𝑆𝐻𝑚𝑎𝑥 + 𝑆ℎ𝑚𝑖𝑛)] + (𝐹1 + 3𝐹3) ∙ (𝑆𝐻𝑚𝑎𝑥 − 𝑆ℎ𝑚𝑖𝑛) − 2𝐹4 ∙
(𝑃𝑤 − 𝑃𝑝) (5.9)
𝐹1 =1
√𝜋𝑎∫ 𝐺(𝑟)𝑑𝑟
𝐿
𝑅 (5.10)
𝐹2 =1
√𝜋𝑎∫
𝑅2
𝑟2 𝐺(𝑟)𝑑𝑟𝐿
𝑅 (5.11)
𝐹3 =1
√𝜋𝑎∫
𝑅4
𝑟4 𝐺(𝑟)𝑑𝑟𝐿
𝑅 (5.12)
𝐹4 =1
√𝜋𝑎∫ 𝐺(𝑟)𝑑𝑟
𝐿
𝐷 (5.13)
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𝐺(𝑟) =1.3−0.3(
𝑟−𝑅
𝑎)
5 4⁄
√1−(𝑟−𝑅
𝑎)
2 (5.14)
where 𝐹1, 𝐹2, 𝐹3, and 𝐹4 are integral functions, which solely depend on the geometry of
the wellbore-fracture system, i.e. wellbore radius, fracture length, and LCM bridge
location. They are, therefore, referred to as geometry terms in this study.
5.2.1.5 Fracture Pressure
Once the fracture-tip stress intensity factor in Eq. 5.9 is found, the fracture pressure
can be determined based on a linear-elastic fracture mechanics criterion. In this study, an
open mode linear-elastic fracture criterion is used, which assumes that fracture initiation
occurs once the stress intensity factor at the fracture tip reaches the fracture toughness, i.e.
𝐾𝐼 = 𝐾𝐼𝐶 (5.15)
where 𝐾𝐼𝐶 is the fracture toughness of the formation rock, taken to be a material property
which describes the ability of the rock material to resist fracture growth.
Fracture pressure, therefore, can be obtained by combining Eqs. 5.9 and 5.15:
𝑃𝑓 =1
2∙
1
𝐹1+𝐹2−𝐹4∙ 𝐾𝐼𝐶 +
1
2∙
𝐹1+𝐹2
𝐹1+𝐹2−𝐹4∙ (𝑆𝐻𝑚𝑎𝑥 + 𝑆ℎ𝑚𝑖𝑛) −
1
2∙
𝐹1+3𝐹3
𝐹1+𝐹2−𝐹4∙ (𝑆𝐻𝑚𝑎𝑥 − 𝑆ℎ𝑚𝑖𝑛) −
𝐹4
𝐹1+𝐹2−𝐹4∙ 𝑃𝑝 (5.16)
where 𝑃𝑓 is the fracture pressure of the wellbore after bridging the fractures.
Eq. 5.16 illustrates that fracture pressure of the wellbore with bridged fractures is a
function of (1) load conditions, including maximum and minimum horizontal stresses and
pore pressure, (2) rock property, i.e. fracture toughness, and (3) geometry of the wellbore-
fracture system reflected by the geometry terms 𝐹1~ 𝐹4. Recall that the fracture pressure
of an intact wellbore, however, is usually defined as a function of load and rock property
only (Fjar et al., 2008; Jin et al., 2013; Zoback, 2010), by combing a Kirsch stress solution
and a tensile failure criterion.
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5.2.2 Model Validation
Validity of the proposed model is tested against an available example solution, as
depicted in Figure 5.7. An elastic wellbore with symmetric arrangement of two fractures is
subject to uniform pressure on the wellbore and fracture surfaces and anisotropic far-field
horizontal stresses. The fracture-tip stress intensity factor solution for this problem can be
found by superposing the solutions for two simpler problems given by Tada et al. (1985),
i.e. (1) the elastic wellbore with two symmetric fractures subject only to far-field
anisotropic stresses, and (2) the same wellbore-fracture system subject only to uniform
pressure in the wellbore and fractures. Its fracture-tip stress intensity factor solution is
formulated as:
𝐾𝐼 = 𝑆ℎ𝑚𝑖𝑛√𝜋𝑎𝐹𝜆1(𝑆0) + 𝑃𝑤√𝜋𝑎𝐹𝜆2(𝑆0) (5.17)
where,
𝐹𝜆1(𝑆0) = (1 − 𝜆1)𝑀(𝑆0) + 𝜆1𝑁(𝑆0) (5.18)
𝐹𝜆2(𝑆0) = (1 − 𝜆2)𝑃(𝑆0) + 𝜆2𝑁(𝑆0) (5.19)
𝑀(𝑆0) = 0.5(3 − 𝑆0)[1 + 1.243(1 − 𝑆0)3] (5.20)
𝑁(𝑆0) = 1 + (1 − 𝑆0)[0.5 + 0.743(1 − 𝑆0)2] (5.21)
𝑃(𝑆0) = (1 − 𝑆0)[0.637 + 0.485(1 − 𝑆0)2 + 0.4𝑆02(1 − 𝑆0)] (5.22)
𝑆0 =𝑎
𝑅+𝑎 (5.23)
𝜆1 =𝑆𝐻𝑚𝑎𝑥
𝑆ℎ𝑚𝑖𝑛 (5.24)
𝜆2 =𝑃𝑓𝑙𝑢𝑖𝑑
𝑃𝑤(= 1 𝑓𝑜𝑟 𝑡ℎ𝑖𝑠 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑐𝑎𝑠𝑒) (5.25)
𝐹𝜆1(𝑆0) and 𝐹𝜆2(𝑆0) are functions of geometry and load of the wellbore-crack system;
𝑆0 is the ratio of fracture length to the distance between fracture tip and wellbore center;
𝜆1 is the horizontal stress ratio; 𝜆2 is the ratio of pressure in the fracture to the pressure
in wellbore, equal to 1 in this case due to the uniform pressure distribution in the wellbore
and fracture.
155
Note that, for Eq. 5.17, the pressure in the fracture is assumed to be uniform and
equal to the wellbore pressure. In order to compare the proposed solution in this study
against the available solution, the LCM bridge location in Figure 5.4 is assumed at the tip
of the fracture (i.e. there is no bridge in the fracture) and the pressure in fracture is equal
to the wellbore pressure due to the small fracture length assumption.
Figure 5.7: Wellbore with two symmetric fractures subject to uniform pressure on the
wellbore and fracture surfaces and anisotropic far-field horizontal stresses.
Input data shown in Table 5.1 were used for model validation. Figure 5.8 shows the
comparison of fracture-tip stress intensity factor between the solution herein and that of
Tada et al. (1985), for different fracture lengths. The two solutions are almost identical
when fracture length is less than or equal to wellbore radius. However, with an increase of
fracture length, the two solutions gradually diverge. But the slight mismatch (less than 10%
relative error) between the two solutions with a fracture less than 3 times that of the
wellbore radius is considered acceptable in this study. It should be noted again that the
proposed analytical solution is limited to predicting fracture-tip stress intensity factor and
156
fracture pressure for a wellbore with small pre-existing fractures, a likely situation in
drilling operations.
Table 5.1: Input parameters for model validation.
Parameter Unit Value
Wellbore radius (𝑅) inch 6
Fracture length (𝑎) inch 1~18
Minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) psi 3000
Maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) psi 3600
Wellbore pressure (𝑃𝑤) psi 4000
Figure 5.8: Comparison of fracture tip stress intensity factors between the proposed model
and Tada’s model (Tada et al., 1985).
5.2.3 Results of Analytical Study
Results of the proposed analytical solution for remedial wellbore strengthening are
presented in this section. As mentioned earlier, the effects of some wellbore strengthening
parameters are not fully understood and justify further elucidation. This includes in-situ
157
stress anisotropy, LCM bridge location, and pore pressure and considerations for field
applications.
Input data for analyses are given in Table 5.2. For illustration purposes, a fracture
length of 6 inches is used here, a value utilized in several previous studies (Alberty and
McLean, 2004; Arlanoglu et al., 2014; Feng et al., 2015a; Wang et al., 2009, 2007b). In
the sensitivity analyses, fracture-tip stress intensity factor and fracture pressure are
calculated for different in-situ stress anisotropies (i.e. 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛⁄ ranging from 1 to 2
with a constant 𝑆ℎ𝑚𝑖𝑛 of 3000psi) and various bridge locations from the fracture mouth
all the way to the fracture tip. The bridge location at the fracture mouth means that no fluid
can penetrate into the fracture, hence the length of invaded zone is zero. As bridge location
moves to the fracture tip, it becomes identical to the situation without facture
bridging/strengthening in which case fluid can penetrate into the full length of the fracture,
all the way to the fracture tip.
Table 5.2: Base input parameters used in the proposed model.
Parameter Unit Value
Wellbore radius (𝑅) inch 6
Fracture length (𝑎) inch 6
Minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) psi 3000
Maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) psi 3600
Wellbore pressure (𝑃𝑤) psi 4000
Pore pressure (𝑃𝑝) psi 1800
Fracture toughness (𝐾𝐼𝐶) psi-in0.5 2000
The effects of in-situ stress anisotropy and bridge location on fracture-tip stress
intensity factor and fracture pressure are shown in Figures 5.9 and 5.10, respectively. They
are calculated according to Eqs. 5.9 and 5.16. The results show that, without bridging the
158
fracture (i.e. the ratio of invaded zone length to fracture length equals to 1), the fracture
has the largest fracture-tip stress intensity factor and the wellbore has the smallest fracture
pressure, meaning that it is very easy for the fracture to initiate and, therefore, lost
circulation to occur. After bridging the fracture, however, the stress intensity factor
decreases and fracture pressure increases significantly, which demonstrates that the
wellbore is effectively strengthened. In addition, with decrease of the invaded zone length
(i.e. bridge location moving closer to the fracture mouth), fracture pressure increases
quickly, hence better strengthening applications. The best location to bridge the fractures
for strengthening a wellbore is at the fracture mouth, near the wellbore wall. Figures 5.9
and 5.10 also show that, at same bridge location, the larger the in-situ stress anisotropy, the
larger the fracture-tip stress intensity factor and the smaller the fracture pressure, meaning
it’s more likely for fracture initiation and lost circulation to occur. Another observation
from Figure 5.10 is that the fracture pressure, with the same bridge location, experiences a
relatively larger enhancement for a wellbore with smaller in-situ stress anisotropy than that
with a larger stress anisotropy. For example, compared with the situation without fracture
bridging, the fracture pressure has an increase of 3400 psi and 2800 psi when bridging at
the fracture mouth for the in-situ stress anisotropies equal to 1.0 and 2.0, respectively. This
suggests that wellbore strengthening operations should be more effective in formations
with relatively uniform in-situ stresses.
159
Figure 5.9: Fracture-tip stress intensity factor with different horizontal stress anisotropies
and bridge locations.
Figure 5.10: Fracture pressure with different horizontal stress anisotropies and bridge
locations.
160
Figures 5.11 and 5.12 show fracture-tip stress intensity factor and fracture pressure
with different bridge locations and pore pressure. As for previous conclusions, these two
figures also show that the wellbore can be better strengthened while bridging the fracture
near the fracture mouth. Given the same horizontal in-situ stress, the stress intensity factor
or the fracture pressure has the same value without bridging the fracture (i.e. the ratio of
invaded zone length to fracture length equals to 1) for different pore pressure as shown in
Figures 5.11 and 5.12. After bridging the fracture at the same location, the wellbore-
fracture system experiences a larger decrease in fracture-tip stress intensity factor and
increase in fracture pressure for the cases with lower pore pressure to minimum horizontal
stress ratio ( 𝑃𝑝 𝑆ℎ𝑖𝑚𝑛⁄ ) than that with higher 𝑃𝑝 𝑆ℎ𝑖𝑚𝑛⁄ ratio. As mentioned in the
introduction, lost circulation is commonly encountered in pressure depleted reservoirs and
deepwater high pressure formations; both of them usually have a very narrow mud weight
window. The results shown in Figures 5.11 and 5.12 suggests that wellbore strengthening
operations should be more effective in depleted zones as compared to deepwater high
pressure zones, because depleted zones usually have relatively small 𝑃𝑝 𝑆ℎ𝑖𝑚𝑛⁄ values
caused by production-induced pore pressure drop. To the contrary, deepwater high pressure
formations usually have relatively large 𝑃𝑝 𝑆ℎ𝑖𝑚𝑛⁄ values because of high formation
pressure and relatively low in-situ stress.
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Figure 5.11: Fracture-tip stress intensity factor with different pore pressure and bridge
locations.
Figure 5.12: Fracture pressure with different pore pressure and bridge locations.
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Figure 5.13 shows the effect of fracture toughness on predicted fracture pressure.
As described above, fracture pressure is calculated with fracture-tip stress intensity factor
equal to fracture toughness. With the same bridge location, a fracture with larger toughness
requires higher pressure to initiate.
Figure 5.13: Fracture pressure with different fracture toughness of the formation rock.
It should be noted that sensitivity analyses presented above in Figures 5.9 through
5.13 are conducted using the base input parameters in Table 5.2. Changing base parameters
only changes the specific values of fracture-tip stress intensity factor and wellbore fracture
pressure, but does not change the trends for effects of horizontal stress anisotropy, bridging
location, pore pressure, and fracture toughness. For example, with a constant minimum
horizontal stress of 3000 psi, it can be observed from Figure 5.10 that a larger in-situ stress
anisotropy leads to a smaller fracture pressure. If the minimum horizontal stress is
increased to 6000 psi, the effect of horizontal stress anisotropy on fracture pressure will
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have the same trend, but a higher fracture pressure value is expected due to the higher in-
situ stress magnitude.
5.2.4 Conclusions and Implications of the Analytical Study
This section presents a fracture-mechanics-based analytical solution for remedial
wellbore strengthening applications. The proposed solution provides an accurate prediction
of fracture pressure enhancement due to fracture bridging, taking into account near-
wellbore stress concentration, in-situ stress anisotropy, and different LCM bridge
locations. The equations presented in this study can be used to perform quick and
quantitative parameter sensitivity analyses to illustrate the undying mechanisms of
wellbore strengthening. In addition, compared with numerical methods, conciseness of the
analytical approach in this study offers the possibility to conduct fracture pressure
prediction in near-real-time drilling operations. Such capability would lead to timely
wellbore strengthening evaluation and mud weight adjustments. Effects of several essential
parameters on wellbore strengthening are assessed using the proposed solution. Based on
parameter sensitivity analyses, the following conclusions and implications are reached:
Fracture pressure can be significantly increased by bridging the small pre-existing
fractures emanating from the wellbore wall in remedial wellbore strengthening
operations. Compared with other loss control methods, which are usually expensive
and require additional equipment, such as using managed pressure drilling or dual
gradient drilling methods to accurately control downhole pressure, along with
casing drilling technology or setting additional casing strings to isolate loss zones,
wellbore strengthening is usually a more economical way to solve lost circulation
problems.
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The closer the bridging location to the fracture mouth, the better strengthening can
be achieved. This means for better application of wellbore strengthening
techniques, it is important to accurately predict fracture geometry, especially
fracture mouth opening, for selecting the best LCM size. Fracture geometry on
wellbore wall during drilling processes is usually unknown. To some degree,
numerical simulation or imaging logging methods can be used to predict or measure
fracture geometry. However, improved or new techniques are needed for acquiring
better knowledge of drilling-induced or pre-exiting natural fractures on the
wellbore wall.
Higher in-situ stress anisotropy results in relatively lower fracture pressure for a
wellbore with pre-existing fractures. Therefore, in drilling under such situations,
close attention should be paid to detect lost circulation events. Wellbore
strengthening methods may be applied to prevent or remedy fluid loss.
Remedial wellbore strengthening applications by bridging pre-existing fractures are
more effective for formations with small in-situ stress anisotropy than those with
large stress anisotropy.
Remedial Wellbore strengthening applications are more effective for formations
with low pore pressure and in-situ stress ratio, such as pressure depleted reservoirs,
as compared to formations with large pore pressure and in-situ stress ratio, such as
deepwater high pressure formations.
5.3 A FINITE-ELEMENT MODEL FOR REMEDIAL WELLBORE STRENGTHENING
APPLICATIONS
While the analytical model proposed in the above section provides a fast procedure
to predict fracture pressure change before and after fracture bridging, it cannot provide
detailed stress and displacement information local to wellbore and fracture in remedial
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wellbore strengthening treatments. Besides, it does not take into account the
poromechanical effect of the formation rock. However, numerical approaches, such as the
finite-element method, can be applied to obtain the detailed information about the evolution
of local stress and deformation in wellbore strengthening operations.
Wang et al. (2009, 2007b) proposed a 2D boundary-element model to simulate two
symmetric fractures on the wellbore wall under anisotropic in-situ stresses, and to obtain
the stress distribution and fracture width before and after bridging the fracture. Guo et al.
(2011) used a 2D finite-element method to investigate fracture width distribution for two
pre-existing fractures, symmetrically located at the wellbore wall under various in-situ
stresses and fracture length. However, the simulation gave no information about fracture
behavior after fracture bridging. Alberty and McLean (2004) employed a 2D finite-element
model to simulate fracture width distribution and hoop stress field after bridging the
fracture, but they only studied cases under nearly isotropic in-situ stresses and bridging
near fracture mouth. What’s more, in all these previous numerical studies, it was assumed
that the rock is linearly elastic. Thus the effects of fluid flow inside the rock and fluid leak-
off through the wellbore wall and fracture surfaces were not considered.
To better simulate remedial wellbore strengthening treatment, a poroelastic finite-
element numerical model, considering fluid flow and fluid leak-off is developed in this
chapter. And then a comprehensive parametric study for remedial wellbore strengthening
based on bridging lost circulation fractures is performed using the proposed numerical
model. The finite-element numerical model discussed in this chapter quantifies near-
wellbore stress and fracture geometry, before and after bridging fractures. Effects of
various parameters are investigated in the parametric study, and the results lead to a number
of useful implications for field applications.
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5.3.1 Numerical Model Development
As mentioned above, linearly elastic rock behavior has been utilized in most
previous studies. The effects of pore pressure and fluid leak-off have not been considered.
The goal of this numerical study is to develop a poroelastic model, taking into account fluid
flow inside the rock and fluid leak-off across fracture faces and the wellbore wall, and to
investigate near-wellbore hoop stress and fracture geometry before and after applying
wellbore strengthening operations. Although the numerical model cannot directly provide
a solution for maximum sustainable pressure of the wellbore in wellbore strengthening
analysis, there is no doubt that, from the points of view of both continuum mechanics and
fracture mechanics, increasing hoop stress – and, hence, stress acting on closing the
fracture – will facilitate the prevention of fracture growth, thus achieving wellbore
strengthening. The link between enhancing hoop stress and increasing maximum
sustainable pressure of the wellbore has also been discussed in detail in a series of papers
(Alberty and McLean, 2004; Aston et al., 2007, 2004a; Dupriest, 2005; Dupriest et al.,
2008; Feng et al., 2015a).
5.3.1.1 Model Geometry
Remedial wellbore strengthening treatments for a vertical wellbore is considered.
The wellbore is assumed to be in a plane-strain condition. Owing to symmetry, only one
quarter of the wellbore is used in the finite-element numerical analysis, as shown in Figure
5.14. Wellbore radius is 4.25 inches. The length and width of the quarter model are 40.25
inches.
A pre-existing fracture is assumed on the wellbore as shown in Figure 5.14, with
its face aligned with the maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥 ) or X-axis direction. The
fracture length of 6 inches is used so as to be consistent with previous work by other
investigators (Alberty and McLean, 2004; Wang et al., 2009, 2007a, 2007b). The fracture
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is plugged with LCMs as shown in Figure 5.14. Through this section, the plug formed
inside the fracture by LCMs across the fracture width is called an LCM bridge. It should
be noted that, although whether LCM bridge material properties (e.g. strength and
permeability) are important or not for wellbore strengthening has been one of the most
debatable questions in the drilling industry, the effect of those material properties is beyond
the study scope of this chapter. The LCM bridge is assumed to be a “perfect” bridge in this
study, which is a rigid body with zero permeability. So there is no fluid flow across the
bridge. To create the effect of bridging the fracture, the velocity in the Y direction at the
bridging location is set equal to zero. As illustrated in Figure 5.14, the quadrant angle is 0
in the X-axis, and increases to 90 degrees around the quarter wellbore.
Figure 5.14: The remedial wellbore strengthening model. Left: geometry and boundary
conditions of the model; Rigth: detailed fracture process zone.
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5.3.1.2 Boundary Conditions
As shown in Figure 5.14, symmetric boundary conditions are applied to the left and
top boundaries of the model. The maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) along the X-direction
and minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) along the Y-direction are applied to the right and
bottom outside boundaries, respectively. Various horizontal stress values are used as shown
in Table 5.3 for investigating the effect horizontal stress contrast.
Wellbore pressure is applied to the inner wall of the wellbore. Pressure in the
fracture is equal to wellbore pressure before bridging the fracture with LCM. After bridging
the fracture, pressure in the fracture ahead of the LCM bridge (from wellbore to the bridge)
is still equal to wellbore pressure. However, pressure behind the LCM bridge (from the
bridge to fracture tip) is set equal to pore pressure, because the pressure in this region will
drop to formation pore pressure with fluid leaking off into the formation and no continuous
fluid supply from wellbore due to the impermeable bridge. Fluid leak-off velocities are
applied on the wellbore wall and fracture faces to simulate fluid leak-off. Various fluid
leak-off velocities and rock properties are also used in the simulations.
5.3.1.3 Input Parameters
Table 5.3 provides the input parameters for the finite-element numerical
simulations.
Total size of the model is about ten times the wellbore size in order to eliminate
boundary effects on near-wellbore stress and strain states.
Formation rock properties are selected for a typical sandstone.
Maximum and minimum horizontal stresses with different stress contrasts, from
1 (𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛 = 1⁄ ) to 1.5 (𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛 = 1.5⁄ ) are used.
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Different pressures behind the bridge 𝑃𝑓𝑏 are used to simulate the sealing
capacity of the LCM bridge, from complete sealing (𝑃𝑓𝑏 = 𝑃𝑝 = 1800 𝑝𝑠𝑖) to
no sealing (𝑃𝑓𝑏 = 𝑃𝑝 = 40000 𝑝𝑠𝑖).
Various Young’s modulus, Poisson’s ratios, LCM bridge locations and leak-off
rates are also selected for parametric sensitivity studies.
Table 5.3: Input parameters for the finite-element model.
Parameter Values Units
Model length 40.25 inches
Model width 40.25 inches
Wellbore radius (𝑅) 4.25 inches
Young’s modulus (𝐸) 1×106, 2×106 psi
Poisson’s ratio (υ) 0.2, 0.3
Minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) 3000 psi
Maximum horizontal stress (𝑆𝐻𝑚𝑎𝑥) 1~1.5·Smin psi
Wellbore pressure (𝑃𝑤) 4000 psi
Pressure in fracture before bridging (𝑃𝑓𝑜) 4000 psi
Pressure ahead of bridge after bridging (𝑃𝑓𝑎) 4000 psi
Pressure behind of bridge after bridging
(𝑃𝑓𝑏)
1800 ~4000 psi
Fracture length (a) 6 inches
Initial pore pressure (Pp) 1800 psi
Leak-off rate 1.0, 2.0, 3.0, 4.0 in/min
Permeability 0.0023 in/min
Void ratio 0.3
LCM bridge location away from wellbore 0.5, 2.0, 3.5, 5.0 inches
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5.3.2 Model Validation
In order to verify the accuracy of the proposed numerical model, existing analytical
solutions are used for validation purpose in this section. The main objective of this
numerical study is to analyze the stress distribution around the wellbore in remedial
wellbore strengthening treatments. However, to the author’s knowledge, there are no
analytical models which can be readily implemented for predicting near-wellbore stress
with fractures emanating from the wellbore wall. Fortunately, several analytical solutions
for stress intensity factor for this problem are available in the literature. Therefore, in this
section, the stress intensity factors calculated from numerical model are validated against
the results of analytical solutions. First, stress intensity factors of an unbridged fracture
with constant length are compared; and then results of a bridged fracture with different
bridging locations are calculated and compared.
5.3.2.1 Stress Intensity Factor of Unbridged Fracture
As described in Section 5.2.2, the solution of stress intensity factor at fracture tip
for the problem of two fractures emanating symmetrically from a pressurized borehole in
an elastic media is given by Tada et al. (1985). For the problem with two unbridged
fractures in the direction of maximum horizontal stress as depicted in Figure 5.7, the stress
intensity factor at both fracture tips can be calculated using Eq. 5.17 according to their
solution.
With the same wellbore and fracture dimensions as given in Figure 5.14 and Table
5.3, wellbore pressure of 4000 psi, and the maximum and minimum horizontal stresses of
3600 psi and 3000 psi respectively, the values of stress intensity factor obtained from the
analytical model and the proposal numerical model without considering porous fluid flow
are shown in Figure 5.15. A comparison between them shows that the results are in good
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agreement with a relative error less than 5%, serving to verify reliability of the proposed
model.
Figure 5.15: Comparison of stress intensity factor for unbridged fracture.
5.3.2.2 Stress Intensity Factor of Bridged Fracture
The proposed numerical model is further validated by comparing against the
analytical model developed in Section 5.2 for a wellbore with bridged fractures. The
problem geometry is very similar to the above validation example. The only difference is
that an LCM bridge is assigned in each fracture wing as shown in Figure 5.4. As assumed
for the numerical model, fluid pressure in the fracture ahead of the LCM bridge (from
wellbore to the bridge) and behind the LCM bridge (from the bridge to the fracture tip) are
also set equal to wellbore pressure and pore pressure, respectively. The analytical solution
of stress intensity factor is given by Eq.5.9. Further details about the development of the
solution can be found in Section 5.2 and in Feng and Gray (2016b).
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Stress intensity factors are calculated and compared using the analytical solution
and the numerical model for the problem with various LCM bridge locations. The
maximum and minimum horizontal stresses are equal to 3600 psi and 3000 psi,
respectively; wellbore pressure is 4000 psi; and pore pressure is 1800 psi. It should be
noted that even though pore pressure is included here, fluid leak-off through the wellbore
and fracture faces is not considered in this validation study. Figure 5.16 shows the
comparison between the two approaches. The good match between the results further
demonstrates the validity of the proposed numerical model.
Figure 5.16: Comparison of stress intensity factor for bridged fracture with various
bridging locations.
5.3.3 Simulation Results and Parametric Analysis
In this section, results from finite-element numerical simulations using the input
parameter in Table 5.3 are presented. Hoop stress around the wellbore and along the
fracture, and fracture width are investigated utilizing the list of influential parameters.
These parameters include horizontal stress contrast, LCM bridge location, fluid leak-off
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rate, pressure behind LCM bridge, Young’s modulus and Poisson’s ratio of the rock
formation.
5.3.3.1 Hoop Stress
Using the finite-element model described above, hoop stress states in the vicinity
of the wellbore and the fracture are analyzed for various combinations of influential
parameters. Figure 5.17 shows the hoop stress state before and after bridging the fracture
with LCM. The bridge location is 2 inches away from the fracture mouth. Throughout this
section, negative and positive stress values mean compressive and tensile stresses,
respectively. It is clear that, before bridging, the fracture tip is under tension and near-
wellbore rock is under compression. However, after bridging the fracture, there is a
compressive stress increase area near the bridging location, meaning the fracture is more
difficult to open; whereas the tensile stress near the fracture tip decreases, meaning the
fracture is more difficult to propagate. After bridging the fracture in wellbore
strengthening, lost circulation is less likely to continue.
Figure 5.17: Hoop stress distribution before (left) and after (right) bridging the fracture in
remedial wellbore strengthening treatment.
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Hoop Stress on Wellbore Wall
Horizontal Stress Contrast 𝑺𝑯𝒎𝒂𝒙/𝑺𝒉𝒎𝒊𝒏. Hoop stress on the wellbore wall from
0 to 90 degrees, with different horizontal stress contrasts, for scenarios without fracture,
with unbridged fracture, and with bridged fracture, are computed and shown in Figures
5.18a, 5. 18b and 5. 18c, respectively. For wellbore without fracture, hoop stress on the
wellbore within 30 degrees to X-axis (𝑆𝐻𝑚𝑎𝑥 direction) is tension. With increase in
horizontal stress contrast, the tensile stress increases. When there is a fracture created as
shown in Figure 5.18b, the tensile stress on the wellbore wall close to fracture changes to
compression. Figure 5.18c is the hoop stress after bridging the fracture at 2 inches away
from the wellbore. It’s not easy to see the stress differences between wellbore with
unbridged fracture and wellbore with bridged fracture from Figures 5.18b and 5.18c. To
facilitate observation, hoop stresses in a single case with 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛⁄ = 1.3 before and
after bridging the fracture are compared in Figure 5.18d. It is clearly shown that
compressive hoop stress on the wellbore wall increases in the area near the fracture mouth
from 0 to 45 degrees and decreases in the area beyond 45 degrees. This is because after
bridging the fracture, fluid pressure behind the bridging point will decrease, as a result the
fracture will try to close. The healing of the fracture will stretch the formation around it,
leading to an increased tension (decreased compression). However, a rigid bridge restrains
healing of the fracture portion near the fracture mouth, resulting in a locally increased
compression; in the far area beyond 45 degrees increased tension still occurs due to the
overall healing behavior of the fracture. Increased compression near the fracture means
that the fracture is less likely to be opened after bridging. Decreased compression beyond
45 degrees means bridging the fracture actually weakens this wellbore portion and new
fractures may generate here with increased wellbore pressure. Figure 5.18d only shows a
single case with 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛⁄ = 1.3 for clarity and conciseness. For other stress
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anisotropies (i.e. 𝑆𝐻𝑚𝑎𝑥 𝑆ℎ𝑚𝑖𝑛⁄ = 1.0, 1.1, 1.2, 1.4 𝑎𝑛𝑑 1.5 ), the same trend with
increased compression near and decreased compression far from the fracture mouth will
also be observed.
Figure 5.18: Hoop stress on wellbore wall for different horizontal stress contrasts: (a)
without fracture, (b) with unbridged fracture, (c) with bridged fracture, and
(d) comparison of hoop stresses before and after bridging for horizontal
stress contrast equal to 1.3.
LCM Bridge Location. Figure 5.19 shows hoop stress on the wellbore wall before
and after bridging the fracture at 0.5, 2 and 5 inches away from wellbore. When the LCM
bridge is close to the wellbore wall, e.g. the 0.5-inch case, there is a dramatic compressive
hoop stress increase on the wellbore wall near the fracture. However, with the bridging
location further away from the fracture, e.g. the 2.0-inch case, there is less hoop stress
increase. For a bridge at 5.0 inch, there is no clear hoop stress change – the stress curve in
Figure 5.19 overlaps with the one before bridging the fracture. Figure 5.19 shows that for
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a bridge at 0.5 inch, at the fracture mouth (0 degree) there is a small compression decrease
compared to the case of no bridging. This unexpected result needs further explanation. One
possible reason is that the stress at 0 degree is read from the corner element at the fracture
mouth in the numerical model; and when the bridge is very close to this element for the
case of bridging at 0.5 inch, some unavoidable computational error in the numerical
simulation might lead to this discrepancy, due to the very large displacement in this
particular corner element. However, the overall result illustrates that, from a hoop stress
enhancement point of view, the best place to bridge the fracture is at the fracture mouth.
Figure 5.19: Hoop stress on wellbore wall for different bridging locations.
Leak-off Rate. Depending on permeability of the rock formation and filter cake on
the wellbore wall, fluid can leak off into the formation at different rates. In this study,
different pre-defined leak-off rates on the wellbore wall and fracture face are applied to
investigate the effect of leak-off. Figures 5.20a, 5.20b, and 5.20c illustrate hoop stress on
the wellbore with different leak-off rates, for scenarios without fracture, with unbridged
fracture, and with bridged fracture, respectively. With no fracture on the wellbore wall,
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there is only a small difference in hoop stresses on wellbore for different leak-off rates, as
shown in Figure 5.20a. However, leak-off rate can have an obvious effect on wellbore hoop
stress when there is an unbridged or bridged fracture, especially in the area near the fracture
mouth, as shown in Figures 5.20b and 5.20c. The higher the leak-off rate, the smaller the
compressive hoop stress on the wellbore wall. Figure 5.20d compares the hoop stress
change on the wellbore wall before and after bridging the fracture with a leak-off rate of 4
inches/s. Similar to Figure 5.18d, there is a compressive hoop stress increase area on the
wellbore wall from 0 to 45 degrees after bridging the fracture. The results indicate that
considering fluid leak-off on the wellbore wall and fracture faces can improve wellbore
strengthening design and evaluation.
Figure 5.20: Hoop stress on wellbore wall for different leak-off rates: (a) without
fracture, (b) with unbridged fracture, (c) with bridged fracture, and (d)
comparison of hoop stresses before and after bridging the fracture for leak-
off rate equal to 4 inches/s.
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Hoop Stress along Fracture Faces
Horizontal Stress Contrast. In remedial wellbore strengthening treatment, it is
desired that hoop stress along the fracture faces can be enhanced, thus increasing the stress
acting to close the fracture. Figures 5.21a, 5.21b and 5.21c show hoop stress along the
facture faces with different horizontal stress contrast for scenarios without fracture, with
unbridged fracture, and with bridged fracture, respectively. The horizontal axis is the
distance away from the wellbore wall along the fracture direction. Before fracture creation,
as shown in Figure 5.21a, hoop stress along the fracture direction is tensile in the near
wellbore region, and becomes compressive with increase of distance away from the
wellbore wall. The higher the horizontal stress contrast, the larger the tensile stress and
tensile area. However, the near-wellbore tensile hoop stress becomes compressive when a
fracture is created, as illustrated in Figure 5.21b; whereas the near-fracture-tip compressive
stress becomes tensile. Figure 5.21b also shows horizontal stress contrast has negligible
influence on hoop stress along fracture faces. Note that in this study, different 𝑆𝐻𝑚𝑎𝑥
values are utilized to change horizontal stress contrast, whereas 𝑆ℎ𝑚𝑖𝑛 is kept as a constant
value. Figure 5.21c is the stress after bridging the fracture. Horizontal stress contrast still
has negligible influence on hoop stress along the fracture. But there is a significant
compression increase near the bridge location at 2 inches away from wellbore wall. Hoop
stresses along fracture, with horizontal stress contrast equal to 1.3, before and after
wellbore strengthening are compared in Figure 5.21d. It is clearly indicated that bridging
the fracture will significantly increase the compressive hoop stress near the LCM bridge
location, which makes the fracture harder to reopen.
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Figure 5.21: Hoop stress along fracture face for different horizontal stress contrasts: (a)
without fracture, (b) with unbridged fracture, (c) with bridged fracture, and
(d) comparison of hoop stresses before and after bridging the fracture for
horizontal stress contrast equal to 1.3.
LCM Bridge Location. Bridge location is a central issue in remedial wellbore
strengthening treatment. Figure 5.22 shows hoop stress along the fracture face with
different bridging locations. When the bridge is very close to the wellbore wall, e.g. the
0.5-inch case Figure 5.22, there is a dramatic compressive hoop stress increase around the
bridge location. However, with the bridge is away from the wellbore wall, the increase of
compressive hoop stress becomes smaller. There is almost no hoop stress change while
bridging the fracture near the fracture tip. For example, for the case with bridge at 5.0
inches away from wellbore wall, the stress after bridging is almost the same as that before
bridging, indicated as two overlapping curves in Figure 5.22. These results also
demonstrate that the best place to bridge the fracture is near the wellbore wall, and it is
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important to determine pre-existing fracture width for designing LCM particle size
distribution and optimizing bridging location.
Figure 5.22: Hoop stress along fracture face for different bridge locations.
Leak-off Rate. Figure 5.23 shows hoop stress along the fracture face with different
fluid leak-off rates from the wellbore wall and fracture faces into the formation. Similarly
to hoop stress around the wellbore (Figures 5.20b and 5.20c), after fracture creation and
bridging, the higher the leak-off rate, the smaller the compressive hoop stress along the
fracture, especially in the region close to the wellbore, as shown in Figures 5.23b and 5.23c.
This result seems completely opposite to the field and experimental observation that
conventional wellbore strengthening techniques are usually more effective in high-
permeable formations (e.g. sandstones) than low-permeable formations (e.g. shales). This
discrepancy can be explained as follows. In field and experimental practices, the
development and quality of the LCM bridge is usually the dominate factor for wellbore
strengthening consequences. In high-permeable formations, larger leak-off rate from the
fracture into the formation can greatly facilitate the consolidation of LCM particles in the
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fracture, so a bridge can develop quickly and usually with high quality. Conversely, in low-
permeable formations, the bridge quality is usually poor due to limited leak-off, resulting
in less strengthening benefit. But in this study, we assume a perfect bridge with infinite
strength and zero permeability exists. From a pure stress analysis, the results in Figures
5.23b and 5.23c are obtained, which only illustrate the stress distribution with a perfect
bridge, but does not reflect the development and quality of the bridge. Investigation of the
effect of LCM properties on wellbore strengthening is beyond the scope of this study.
Figure 5.23d compares hoop stresses along the fracture, before and after bridging the
fracture at 2 inches away from the wellbore, with a leak-off rate of 4 inches/s. Increased
compressive stress and decreased tensile stress are observed around the bridging location
and fracture tip, respectively.
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Figure 5.23: Hoop stress along fracture face for different leak-off rates: (a) without
fracture, (b) with unbridged fracture, (c) with bridged fracture, and (d)
comparison of hoop stresses before and after bridging the fracture for leak-
off rate equal to 4 inches/s.
Pressure Behind LCM Bridge. For the numerical studies above, the bridge is
assumed to be impermeable and therefore prevents pressure communication across the
bridge. It is assumed that pressure behind the LCM bridge will decline to formation
pressure as the fluid leaks off. However, in real situations, the bridge is likely not
completely impermeable in real situations, and fluid can flow across the bridge due to
pressure differential. Depending on fluid penetration across the bridge or the permeability
of the bridge, pressure behind the LCM bridge can vary from formation pressure (perfectly
impermeable bridge) to wellbore pressure (fully permeable bridge). Figure 5.24 shows
hoop stress along the fracture faces for pressure behind the LCM bridge varying from
formation pressure, 1800 psi, to wellbore pressure, 4000 psi. The higher the pressure
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behind the LCM bridge, the smaller the compression near the bridge location and the higher
the tension at the fracture tip. This lessens the effectiveness of wellbore strengthening. The
results in Figure 5.24 illustrate the importance of forming a low permeability bridge.
Figure 5.24: Hoop stress along fracture face for different pressures behind LCM bridge,
varying from formation pressure of 1800 psi to wellbore pressure of 4000
psi.
It should be noted that all of the above hoop stress calculations are performed for
the fracture with a constant length of 6 inches. This length was selected for two reasons.
First, it is argued that 6 inches is an indicative fracture length from practical experience by
Alberty and McLean (2004); and second this value is consistent with several numerical and
analytical wellbore strengthening studies by other investigators (Alberty and McLean,
2004; Arlanoglu, 2012; Mehrabian et al., 2015; Wang et al., 2007a, 2007b). Fracture length
is a major parameter determining the maximum sustainable pressure of the wellbore.
Changing its value will greatly affect wellbore strengthening results. As indicated by
Morita and Fuh (2012), if the fracture is effectively bridged at the same distance from the
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bridging point to the fracture mouth, the larger the total length of the fracture, the better
strengthening result can be achieved. More details for wellbore strengthening with different
fracture lengths can be found in the analytical studies by Morita and Fuh (2012) and Shahri
et al. (2014).
5.3.3.2 Fracture Width
Knowledge of fracture geometry, especially fracture width distribution, is essential
for selecting LCM sizes in designing a wellbore strengthening treatment. In this section,
the effects of a number of parameters on fracture width distribution before and after
bridging the fracture are investigated.
Figure 5.25 shows vertical displacement perpendicular to the fracture face, before
and after bridging the fracture using LCM. The bridge location is 2 inches away from the
wellbore wall. The displacement magnitude along the fracture face is the half width of the
fracture. The minus sign in Figure 5.25 means the fracture opening displacement is
opposite to the direction of the Y-axis. The blue and red colors indicate larger and smaller
fracture opening displacement (or fracture width), respectively. Fracture width decreases
after bridging the fracture, especially in the region behind the bridge location.
Note that in the following figures for plotting fracture width, a minus sign is still
used. But this sign only means the direction of fracture opening displacement, not a
negative fracture width.
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Figure 5.25: Vertical displacement distribution in the model before and after bridging the
fracture in wellbore strengthening.
Horizontal Stress Contrast. Figures 5.26a and 5.26b show fracture half-width
distribution along the fracture length with different horizontal stress contrasts with
unbridged and bridged fracture, respectively. The bridge location is 2 inches away from
the wellbore. For both cases, with increase in horizontal stress contrast, fracture half-width
increases, especially in the area close to the wellbore. Figure 5.26b shows that horizontal
stress contrast has a very small effect on fracture width behind the LCM bridge after
strengthening. Fracture widths before and after bridging the fracture for a particular case
with horizontal stress contrast equal to 1.3 are compared in Figure 5.26c. The results shows
that the fracture width behind the LCM bridge has a significant decrease after bridging the
fracture, which means the fracture is trying to close after the strengthening operation.
However, fracture width experiences a much smaller decrease ahead of bridge location,
likely due to the relatively higher fluid pressure in this part of the fracture.
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Figure 5.26: Fracture half-width distribution for different horizontal stress contrasts: a)
before bridging the fracture, b) after bridging the fracture, and c)
comparison of fracture half-widths before and after bridging the fracture for
horizontal stress contrast equal to 1.3.
LCM Bridge Location. Fracture half-widths before bridging the fracture and after
bridging it at 0.5, 2.0, 3.5 and 5.0 inches away from the wellbore wall are computed and
shown in Figure 5.27. When the fracture is bridged near the wellbore wall, for example the
0.5-inch case, the fracture experiences a larger decrease in its width, compared with
bridging away from the wellbore. For a bridge at 5.0 inch, there is no clear width change –
the width curve in Figure 5.27 overlaps with that before bridging the fracture. Since the
objective of wellbore strengthening is to prevent fracture opening and propagation, Figure
5.27 also shows that the best place to bridge the fracture is at the fracture mouth.
Figure 5.27: Fracture half-width distribution for different LCM bridge locations.
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Leak-off Rate. Leak-off rate also affects fracture width distribution. Figures 5.28a
and 5.28b are fracture half-widths for different leak-off rates before and after bridging the
fracture at 2 inches away from wellbore. For both cases, higher leak-off rate results in
smaller fracture width. Figure 5.28b illustrates that leak-off rate has a relatively larger
influence on fracture width ahead of the LCM bridge location, with a relatively smaller
influence behind the bridge.
Figure 5.28: Fracture half-width distribution for different leak-off rates: (a) before
bridging the fracture, (b) after bridging the fracture.
Young’s Modulus. Young’s modulus has a significant influence on fracture width.
As shown in Figure 5.29, larger Young’s modulus leads to a smaller width for the fracture,
both before and after bridging at 2 inches away from the wellbore. Additionally, a fracture
with smaller Young’s modulus has a much larger width decrease after bridging than a
fracture with larger Young’s modulus, especially the part behind the LCM bridge. This
result demonstrates the importance of accurate Young’s modulus data for reliable
prediction of fracture width, hence LCM size optimization.
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Figure 5.29: Fracture half-width distribution for different Young’s Modulus before and
after bridging the fracture.
Poisson’s Ratio. Figure 5.30 shows fracture half-width before and after bridging
the fracture at 2 inches away from the wellbore wall with Poisson’s ratios of 0.2 and 0.3.
The results show, compared with Young’s modulus, Poisson’s ration has a much smaller
influence on fracture widths before and after bridging. An apparently larger Poisson’s ratio
only results in a slightly smaller fracture width.
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Figure 5.30: Fracture half-width distribution for different Poisson’s ratios before and after
bridging the fracture.
Pressure behind LCM Bridge. As discussed above, pressure behind the LCM
bridge can vary from formation pressure (perfectly impermeable bridge) to wellbore
pressure (fully permeable bridge) depending on the permeability of the LCM bridge. Figure
5.31 shows the fracture half-width distribution for pressure behind the LCM bridge varying
from formation pressure, 1800 psi, to wellbore pressure, 4000 psi. The lower the pressure
behind the LCM bridge, the smaller the fracture width behind the LCM bridge. This, again,
means the less permeable the LCM bridge, the more effective the strengthening operation.
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Figure 5.31: Fracture half-width distribution for different pressure behind LCM bridge,
varying from formation pressure 1800 psi to wellbore pressure 4000 psi.
5.3.4 Conclusions and Implications of the Numerical Study
For effectively strengthening a wellbore, it is necessary to understand the
fundamentals of lost circulation and wellbore strengthening. Because the fracture
growth/closure behavior and corresponding stress and dimension changes are difficult to
measure in the field, numerical simulation (e.g. finite element modeling) is especially
useful for studying these processes and revealing useful information for field applications.
Regarding hoop stress as an important consideration in remedial wellbore strengthening
treatment, the following conclusions and field implications are indicated from the above
finite-element modeling analysis.
After bridging a fracture, there is a compression increase area near the bridging
location, whereas the tensile stress near the fracture tip decreases. This means that
the fracture becomes more difficult to reopen and propagate, and lost circulation is
less likely to continue.
Hoop stress around wellbore can be substantially increased by bridging the fracture
near the wellbore wall. Bridging the fracture near the fracture tip does little to
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increase hoop stress and strengthen the wellbore. It is important to correctly predict
fracture width and optimize LCM size distribution in wellbore strengthening
design.
Fluid leak-off affects both hoop stress and fracture width distributions. It is
important to consider fluid leak-off on the wellbore wall and fracture faces in
predicting fracture geometry, optimizing LCM size distribution, and evaluating
potential hoop stress enhancement with wellbore strengthening.
For effectively strengthening the wellbore by preventing fluid communication
between the wellbore and fracture tip, a desirable LCM bridge should have low
enough permeability to ensure there is no fluid flow across the bridge and the
fracture portion behind the bridge can close due to fluid leak-off. The permeability
of the bridging plug is likely a more important parameter than its strength.
Young’s modulus has a significant influence on fracture width distribution in
wellbore strengthening. It is critical to obtain reliable Young’s modulus values for
prediction of fracture width, hence optimization of LCM size. However, Poisson’s
ratio only has a very small effect on fracture width distribution.
It is well-known from fracture mechanics that the longer the fracture the less net
pressure (difference between fluid pressure inside the fracture and field stress acting
on fracture faces) is needed to propagate the fracture. However, in a real field
situation, the fracture length is hard to predict. However, from these simulation
results the best place to bridge the fracture, from a point of view of hoop stress
enhancement, is at its mouth or entrance on the wellbore wall. This simplifies the
problem, i.e., bridge the fracture mouth, then fracture length is of much less
consequence.
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5.4 SUMMARY
The goal of this chapter is to develop models to investigate the remedial wellbore
strengthening method based on plugging/bridging lost circulation fractures using LCMs.
An analytical solution and a finite-element model are proposed for modeling this problem.
The analytical solution derived based on liner elastic fracture mechanics and
superposition principle can be used for a fast prediction of fracture pressure enhancement
due to fracture bridging, taking into account near-wellbore stress concentration, in-situ
stress anisotropy, and different LCM bridge locations. The equations presented in the
solution can be used to perform quantitative parameter sensitivity analyses to illustrate the
undying mechanisms of wellbore strengthening. In addition, compared with numerical
methods, conciseness of the analytical approach offers the possibility to conduct fracture
pressure prediction in near-real-time drilling operations. Such capability would lead to
timely wellbore strengthening evaluation and mud weight adjustments.
However, the analytical model cannot provide detailed stress and displacement
information local to wellbore and fracture in remedial wellbore strengthening treatments.
Besides, it does not take into account the porous nature of formation rock. Therefore, a
poroelastic finite-element numerical model, considering fluid flow and fluid leak-off, is
also developed in this chapter. The finite-element model can be used to quantify near-
wellbore stress and fracture geometry, before and after bridging fractures in remedial
wellbore strengthening treatment. Effects of various parameters are investigated through a
comprehensive parametric study using the numerical model, and several useful
implications for field applications are proposed based on the parametric study.
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CHAPTER 6: Cement Interface Fracturing3
With ongoing environmental concerns and increasingly stringent regulations, the
traditional topic of wellbore integrity is more important than ever before. A great deal of
importance is attached to the cement sheath because it is necessary to provide zonal
isolation and well integrity during the life of a well. A quality cement sheath with low
permeability can effectively prevent hydraulic communication of formations to the
borehole and to other formation layers cased off during well construction. However,
leakage from the cement sheath frequently leads to well integrity problems in operations.
These failures may occur at any time throughout the life of a well - during drilling,
completion, production, and even after well abandonment. Debonding at the casing/cement
or cement/formation interfaces, which may result in substantial flow channels and fluid
leakage, is often responsible for the loss of wellbore integrity.
In this chapter, a three-dimensional finite-element framework is developed to exam
the possibility of fluid leakage through casing shoe and along the weak cement interface
when there is pressure buildup in the wellbore due to change of drilling or completion fluid,
conduction of injectivity tests, and etc. Cement interfaces to casing and formation are
represented with zero-thickness pore pressure cohesive element layers, so as to model
debonding fracture initiation and propagation, as well as tangential and normal fluid flow
within the fracture. The model is used to quantify the length, width, and circumferential
coverage of the debonding fracture.
3 Parts of this chapter has been published in the following conference paper which was supervised by K. E.
Gray:
Feng Y., Podnos, E. and Gray K. E. “Well Integrity Analysis: 3D Numerical Modeling of Cement
Interface Debonding" presented at the 50th US Rock Mechanics / Geomechanics Symposium held in
Houston, Texas, USA, 26-29 June 2016.
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6.1 INTRODUCTION
During well construction, fluid may not only lose in the open-hole section through
hydraulically induced fractures, it may also lose through leakage at casing shoe with poor
cement integrity.
Well integrity, always of importance, is becoming more so because of increasing
environmental concerns and regulatory activities. Moreover, increasingly hostile
environments such as HT/HP fields, ultra-deep-water fields, and geothermal fields, along
with more complicated development operations, such as those encountered in gas
producing wells, gas storage wells, water injectors, and cuttings/waste injectors, all present
new challenges for well integrity. Failure of well integrity can lead to costly remedial
operations or total loss of the well, severe environment contamination, and even fatal
accidents to operating personnel.
The cement sheath is the heart of well integrity. It is expected to ensure well
integrity by providing zonal isolation and support for the casing throughout the life of a
well, from well construction, hydrocarbon production, to post-abandonment. A successful
cementing job is expected to result in complete zonal isolation, without leaving any leakage
pathway in the annulus between casing and formation. Unfortunately, this goal is not
always achieved, and failure of cement sheath commonly occurs during the life of a well
(Fourmaintraux et al., 2005).
Failure modes of the cement sheath may include: (1) tensile cracking in the cement,
(2) plastic deformation in the cement, and (3) debonding at the cement/casing and/or
cement/formation interfaces. These modes are thought to result from high stress/pressure
levels encountered in the cement sheath during a well’s life. Cement sheath failure may
also be induced by improper cement placement because of high well inclination, poor hole
calibration, poor centralization, poor selection of chemical agents, or poor mud removal
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(Bois et al., 2011). For this study, attention is focused mainly on debonding at the cement
interfaces when there is pressure buildup in the wellbore due to change of drilling or
completion fluid, conduction of injectivity tests, and etc.
Several researchers have developed models to simulate cement interface
debonding, based on simplifications such as assumptions of linear elastic materials (casing,
cement, rock) and initially intact cement sheath (Bosma et al., 1999; Fleckenstein et al.,
2001; Fourmaintraux et al., 2005; Gray et al., 2009; Pattillo and Kristiansen, 2002; Ravi et
al., 2002; Shahri et al., 2005). However, laboratory tests performed on a Class-G cement
system showed that cement is better characterized as a porous media, and modeled not as
a one-phase linear elastic medium (Bois et al., 2012; Ghabezloo et al., 2008). It is also well
known that the cement and formation may not always behavior elastically, and plastic
deformation commonly occurs, especially in the vicinity of wellbore where high stress
level usually exists due to local stress concentration (Gray et al., 2009). Therefore, selection
of appropriate constitutive laws, which can correctly describe cement and formation
behavior, is a key point for cement sheath modeling (Bois et al., 2011).
Most previous studies (Bosma et al., 1999; Fleckenstein et al., 2001; Fourmaintraux
et al., 2005; Gray et al., 2009; Ravi et al., 2002; Shahri et al., 2005) did not consider initially
existing defects at the cement interfaces due to poor mud removal, cement shrinkage during
hydration, or other operational factors, before simulating interface debonding by applying
stress/pressure boundary conditions. Additionally, most researchers have modeled
interface debonding as a tensile or shear failure due to high local stress induced by the
combined effects of far-field in-situ stress (usually not isotropic), fluid pressure in wellbore
(likely subjected to large changes during the life of a well), temperature changes, and
different mechanical properties of the casing, cement and formation. However, when an
initial crack due to poor mud removal or cement shrinkage, or an induced crack resulting
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from excessive shear/tensile stress, exists at the cement interface, then fluid may invade
into the crack, leading to pressure buildup and, ultimately to a fluid-driven fracture that
propagates along the interface. Consequences of the debonding fracture propagation, such
as leakage of formation fluid to the surface and/or uncontrolled inter-zonal flow, can result
in severe operational troubles and/or substantial environmental pollution. Circumstances
in which fluid-driven debonding fracture may occur include:
Changing mud while drilling
Pressure testing/cycling during drilling, completion, and production
Perforation operations
Hydraulic fracturing
Fluid injection, such as gas injection, produced water re-injection, drilling
cutting re-injection, steam injection, and flooding operations.
Gray et al. (2009) developed numerical models to investigate cement interface
debonding by considering the casing/cement interface as a contact condition, which may
allow zero or some amount of tension transmission across the interface, corresponding to
the cases with no bonding strength and finite bonding strength respectively. Other
researches modeled the cement interface as a layer of interface elements based on a
Coulomb friction model (Bosma et al., 1999; Ravi et al., 2002). While such models may
be satisfactory for analyzing interface debonding when there is no fluid invasion into the
annular cracks after debonding, they cannot simulate the propagation of fluid-driven cracks
along the interface, which requires fully-coupled modeling of the mechanical behavior of
the casing/cement/formation and fluid flow in the cracks.
Great progress has been made in fully-coupled modeling of fluid-driven fracture in
porous media in the past decade, such as the coupled pore pressure cohesive zone method
and coupled pore pressure extended finite element method (Kostov et al., 2015; Searles et
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al., 2016; Wang, 2015; Yao et al., 2010). These methods can successfully account for
several key phenomena of fluid-driven fracture in porous media, including fluid flow
within the fracture, pore fluid flow in the porous media, deformation of porous medium,
and fracture propagation (Zielonka, et al. 2015). However, these techniques have not
been applied to the study of cement sheath integrity until recently (Wang and Taleghani,
2014). To the author’s knowledge, Wang and Taleghani (2015) first applied a coupled pore
pressure cohesive zone method to this problem. They successfully modeled fluid-driven
fracture propagation along the cement interfaces, which improves the understanding of
cement bond failure due to excessive fluid pressure buildup at the interface.
The study presented in this chapter is another implementation of the coupled pore
pressure cohesive zone method to cement sheath integrity modeling. While the overall
approach of this chapter is similar to that of Wang and Taleghani, i.e. both are based on
cohesive zone theory and finite-element analysis, the model assumptions, interpretation
methods, and specific focus are different. Wang and Taleghani (2015) mainly focused on
investigating the effects of interface properties on the propagation of debonding fracture,
while the aim of this study is to simulate non-uniform debonding fracture with various in-
situ stress conditions, cement and formation properties, and pre-existing cracks at the
cement interfaces.
6.2 DEVELOPMENT OF CEMENT INTERFACE FRACTURE MODEL
6.2.1 Modeling Goals
The first goal of this study is to develop a general finite-element framework for
simulating cement sheath debonding in a vertical well due to fluid-driven fracture
propagation along the circular interfaces of cement to casing and formation. The
framework should also take into account other crucial aspects of the
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casing/cement/formation system, including nonlinearity of the material properties,
interaction between different materials, poromechanical nature of cement, plastic
deformation of cement and formation, three dimensionality and anisotropy of in-situ
stresses. The framework should have the flexibility to be readily modified for a non-vertical
well, casing eccentricity, and more complex material models.
The second objective of the paper is to apply this finite-element framework to
investigate the propagation of debonding fracture from a casing shoe, with various initial
micro-annular-fractures around the shoe, different in-situ stress anisotropies, and different
material properties. Also, development of plastic deformation in the cement during
debonding fracture growth will be followed.
To achieve these goals, the SIMULIA Abaqus® Standard finite-element package
was selected for developing the model. It is a general-purpose finite-element code
developed to solve linear and nonlinear stress-analysis problems, with capabilities of
constructing 3D geometry, a variety of nonlinear solvers, and an extended collection of
material models (Gray et al., 2009). The recently developed capabilities of the Abaqus®
package, in modeling fully-coupled, fluid-driven fracture in porous medium, are
particularly useful for studies of hydraulic fracturing related problems in the petroleum
industry (Kostov et al., 2015; Searles et al., 2016; Wang, 2015; Yao et al., 2010). In this
study, fracture propagation and fluid flow in the fluid-driven fracture along the cement
interface are modeled using coupled pore pressure cohesive zone method as described in
Section 3.2. A traction-separation constitutive law and a fluid flow constitutive law are
incorporated into the cohesive zone model to describe the two phenomena respectively.
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6.2.2 Model Geometry and Discretization
Geometry. Since a debonding fracture may develop both circumferentially and
vertically at the cement interface with a non-uniform pattern, the traditional 2D model,
assuming plane stress/strain conditions, cannot correctly predict fracture growth (Wang
and Taleghani, 2014). In addition, accurate modeling of nonlinear effects, such as interface
discontinuity and material plasticity, depends highly on the ability to model 3D stress state
(Gray et al., 2009). Therefore, a three-dimensional model was developed in this study for
the cement sheath integrity problem, as shown in Figure 6.1.
A vertical well is modeled with its axial direction coinciding with the Z-axis of the
Cartesian coordinate system, as depicted in Figure 6.1. The X-axis and Y-axis are assigned
along the direction of the maximum and minimum horizontal stresses, respectively. The
components of the model include casing, cement, formation as shown in the plane view in
Figure 6.1b, and the interface between casing and cement as shown Figure 6.1c. For the
present study, only debonding at the casing/cement interface is considered, while the outer
interface at the cement/formation is assumed to be functionally bonded. However,
debonding at the outer interface can be easily included with minor model modification.
Because of the symmetry of the problem, only one quarter of the system is modeled.
The casing inner radius is 8.41 cm, and outer radius is 9.69 cm, i.e. the casing
thickness is 1.28 cm. The drilling hole size is 12.07 cm, hence the thickness of the cement
sheath between casing and formation is 2.38 cm. The casing/cement interface is modeled
using a layer of cohesive elements with zero thickness. The total size of the quarter model
is 10×10×40 m. The casing shoe, where leakage occurs and pressure builds up, is at the
bottom of the wellbore.
Discretization. The casing is discretized using 3D linear full-integration elements
without degree of freedom of pore pressure. Cement and formation are discretized using
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3D linear full-integration elements with degree of freedom of pore pressure. The
casing/cement interface is discretized with a layer of coupled pore pressure and
deformation cohesive elements which can model propagation of the debonding fracture
and fluid flow in the fracture, as described in Section 3.2. Figure 6.1 shows the overall
mesh of the model, with 1500, 2000, 20500, and 500 elements being used to mesh casing,
cement, formation, and cohesive interface, respectively.
Since interface failure, large plastic deformation, and high stress levels are expected
close to the wellbore, especially near the casing shoe, the mesh in this region is refined,
while coarser elements are used farther away from the casing shoe, both in lateral and
vertical directions.
Figure 6.1: Cement sheath model. (a) the one-quarter geometry; (b) top view of the
casing/cement/formation system; (c) the interface between casing and
cement.
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6.2.3 Material Variables
Casing. The steel casing is assumed to be a linearly elastic material in this study,
with Young’s modulus, 𝐸, and Poisson’s ratio, 𝑣, equal to 2 × 108 kPa and 0.27 ,
respectively.
Cement and Formation. The cement and formation rock are modeled as
elastic/perfectly plastic porous materials. The Mohr-Coulomb plasticity model is used. The
elastic material parameters include Young’s modulus, 𝐸, and Poisson’s ratio , 𝑣, ; the
Mohr-Coulomb plastic parameters include internal friction angle, 𝜑, and cohesive
strength , 𝑐 ; and the porous properties include porosity, ∅, and permeability, 𝑘 . The
properties of cement and formation used in this study are reported in Tables 6.1 and 6.2,
respectively, with each of them consisting of a base case and several supplementary cases
for sensitivity study purposes.
Interface Bond. As stated above, the interface between casing and cement is
modeled using a layer of zero thickness coupled pore pressure and deformation cohesive
elements, which incorporates a traction-separation law to describe fracture propagation
behavior and a flow rule for fluid flow in the fracture. A major challenge in using a cohesive
model is the determination of cohesive properties of the casing/cement interface. Very little
data are available in the literature. Here the cohesive properties used for the casing/cement
interface, including tensile strength, shear strength, cohesive stiffness, and critical energy,
are based on data reported by Wang and Taleghani (2015), where they estimated cohesive
properties of the cement interface through numerically simulating and matching the pipe
(casing) push-out tests by Carter and Evans (1964). The leakage fluid is assumed to be
water, and a very small leak-off coefficient is used for the fracture surface on the cement
side due to low permeability of the cement, while zero leak-off is defined on the casing
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side due to the impermeable nature of casing. The parameters utilized for the cohesive
interface bond are summarized in Table 6.3.
Table 6.1: Cement properties.
Young’s
Modulus, 𝐸
Poisson’s
Ratio, 𝑣
Friction
Angle, 𝜑
Cohesive
strength, 𝑐
Permeability,
𝑘
Porosity,
∅
Type kPa (o) kPa mD
C1 2.00E+07 0.26 27 8.00E+03 0.05 0.2
C2* 3.00E+07 0.26 27 1.00E+04 0.05 0.2
C3 8.00E+07 0.26 27 1.50E+04 0.05 0.2
Note: C2 is the base case.
Table 6.2: Formation properties.
Young’s
Modulus, 𝐸
Poisson’s
Ratio, 𝑣
Friction
Angle, 𝜑
Cohesive
strength, 𝑐
Permeability,
𝑘
Porosity,
∅
Type kPa (o) kPa mD
F1* 3.30E+06 0.26 30 5.00E+03 1.0 0.2
F2 5.00E+06 0.26 30 7.00E+03 1.0 0.2
F3 4.00E+07 0.26 30 1.20E+04 1.0 0.2
Note: F1 is the base case.
Table 6.3: Interface bond properties.
Cohesive
Stiffness
Shear
Strength
Tensile
Strength
Critical
Energy
Leak-off
Coefficient
Fluid
Viscosity
kPa kPa kPa J/m2 m/s/Pa cp
8.50E+07 2,000 500 100 5.897E-12 1.0
6.2.4 Boundary Conditions and Simulation Steps
Boundary Conditions. A symmetric boundary technique is used to reduce
computation burden of the 3D model, hence only one-quarter of the
casing/cement/formation domain is modeled as shown in Figure 6.1. Symmetry boundary
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conditions are applied to the front and left surfaces of the model in Figure 6.1. The vertical
displacement at the bottom surface is constrained. An overburden pressure of 11,000 kPa
is applied at the top surface of model. Maximum horizontal stress parallel to the X-axis
and minimum horizontal stress parallel to the Y-axis are applied on the right surface and
back surface of Figure 6.1, respectively. Different horizontal stress ratios, as given in Table
6.4, are used to investigate the effect of stress anisotropy on cement interface debonding.
The initial pore pressure is assumed equal to 6,000 kPa. A constant wellbore fluid pressure
of 7,200 kPa is applied on the inner surface of the casing. Because thickness of the
simulation domain is relatively small, no variation in initial stress or pressure is considered
in the vertical direction. Table 6.4 summarizes different types of loads assigned to model.
Table 6.4: In-situ stresses, pore pressure and wellbore pressure applied to the model.
Horizontal
Stress
Ratio*
Minimum
Horizontal
Stress
Maximum
Horizontal
Stress
Overburden
Pressure
Initial
Pore
Pressure
Wellbore
Pressure
Type kPa kPa kPa kPa kPa
SR0* 1 9,600 9,600 11,000 6,000 7,200
SR1 1 8,000 8,000 11,000 6,000 7,200
SR1.1 1.1 8,000 8,800 11,000 6,000 7,200
SR1.2 1.2 8,000 9,600 11,000 6,000 7,200
SR1.25 1.25 8,000 10,000 11,000 6,000 7,200
SR1.3 1.3 8,000 10,400 11,000 6,000 7,200
SR1.35 1.35 8,000 10,800 11,000 6,000 7,200
SR1.4 1.4 8,000 11,200 11,000 6,000 7,200
Note: SR0 is the base case. Horizontal stress ratio is the ratio of maximum horizontal stress to
minimum horizontal stress.
Initially existing cracks, through which fluid leakage occurs and a fracture initiates,
are assigned at the casing/cement interface around the casing shoe. The width and height
of the initial cracks are constant and equal to 2 mm and 20 mm respectively, while various
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circumferential crack extents are used, in an attempt to investigate interface debonding
patterns with different initial crack shapes. The circumferential extent of the initial crack
is quantified using an arc angle. A 0𝑜 arc angle indicates there is no initial crack, and a
90𝑜 angle means a crack extending throughout the circumference of the circular interface.
Table 6.5 provides a detailed description of the initial crack shapes modeled in this study.
Table 6.5: Geometry of the initial cracks at the casing shoe.
Type Q1 Q2 Q3 Q4*
Initial
Crack
Shape
Circumferential
Extent, (o) 30 45 60 90
Thickness, mm 2 2 2 2
Height, mm 20 20 20 20
Note: Q4 is the base case.
Simulation Steps. Two steps were used in each simulation. At the first step, far-
field horizontal stress, overburden pressure, and initial pore pressure are applied to the
mesh. Initial state of stress reaches equilibrium in this step. No debonding occurs during
the equilibrium process except for pre-assigned cracks around the casing shoe. The second
step simulates leakage and pressure buildup at the casing shoe and their consequences,
including debonding fracture propagation along the casing/cement interface and plastic
deformation development in the cement at the same time. Pressure buildup around the
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casing shoe is modeled as a result of fluid charge (leakage) into the cement interface with
a very small charge rate.
6.3 RESULTS AND DISCUSSION
Using the finite-element model described above, the cement interface debonding is
investigated for various combinations of in-situ stresses, pre-exiting cracks around the
casing shoe, and cement and formation properties. The following subsections summarize
the simulation results.
6.3.1 Cement Interface Debonding with Anisotropic Horizontal Stress
In-situ stresses are always compressive and resist opening of cement interface
unless the stress anisotropy is too large. However, with a leakage, fluid pressure may build
up in the interface, ultimately, resulting in sufficient pressure to overcome resistance of the
in-situ compressive stress and, hence, initiation of a hydraulic fracture along the interface.
To investigate the effects of non-uniform (horizontal) in-situ stresses on development of
the debonding fracture, several cases with different horizontal stress ratios as shown in
Table 6.4 are simulated. For comparison purposes, only two cases, SR1 and SR1.2 with
horizontal stress ratios equal to 1.0 (uniform) and 1.2 respectively, are selected for analysis.
Figure 6.2 illustrates creation of the debonding fracture near the casing shoe for the
two selected cases by the end of the second simulation step. A comparison between the two
cases shows that:
For the case (SR1) with uniform horizontal stress, the width of the debonding
fracture, as expected, is uniform around the casing, as shown in Figures 6.2 and 6.3.
This is because everything, i.e. stress and material properties, is uniformly
distributed in this case.
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For the case (SR1.2) with anisotropic horizontal stress, the debonding fracture is
not uniform, with a smaller (or even zero) width in the direction of the maximum
horizontal stress (X-axis) and a larger width in the direction of the minimum
horizontal stress (Y-axis), as shown in Figures 6.2 and 6.3. This is because the
casing/cement interface is under a larger compressive stress in the direction of
maximum horizontal stress, which resists fracture opening, but under a smaller
compression in the direction of minimum horizontal stress. This observation is
consistent with the conclusion of hydraulic fracturing studies that a fracture should
occur in the plane perpendicular to the least principal stress (Gray et al., 2009;
Hubbert and Willis, 1957; Yew and Weng, 2014; Zoback, 2010).
Figure 6.2: Interface fracture opening of two cases with uniform horizontal stress (SR1)
and non-uniform horizontal stress (SR1.2). The pictures are top views of the
cut sections of the casing/cement/interface system at 0.5 m above the casing
shoe.
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Figure 6.3: Interface fracture width around wellbore for the two cases in Figure 6.2. 0-
degree and 90-degree correspond to the directions of the maximum and
minimum horizontal stress, respectively.
6.3.2 Cement Interface Debonding with Different Initial Cracks
Figure 6.4 shows the growth of the fracture, distribution of pressure in the fracture,
and distribution of plastic strain in the cement by the end of the second step for the four
cases with different initially existing cracks around the casing shoe as given in Table 6.5.
The in-situ stress type used is SR1.2 in Table 6.4 and the material properties used are the
base values described in Section 6.2.3. The results show that the patterns of the
developments of fracture, pressure, and plastic strain are consistent for each case, i.e. where
fracture occurs, high pressure in the fracture and large plastic strain in the cement exist.
The large plastic strain induced by fracture propagation may lead to material damage, such
as creation of voids and micro-cracks in the cement (Gray et al., 2009), and consequently
further damage of well integrity.
Another observation is that the fluid-driven debonding fracture tends to develop
vertically, rather than circumferentially around the casing. When a 90𝑜 initial-
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circumferential crack exists, for example, case Q4 in Table 6.5 and Figure 6.4, the fracture
will propagate at the minimum horizontal stress (Y-axis) side because the resistance to
fracture opening is the smallest there as explained above. However, if the initial crack does
not cover the full circumference of the interface, such as cases Q1, Q2, and Q3 in Table
6.5 and Figure 6.4, the fracture is more inclined to propagate in the vertical direction, rather
than in the circumferential direction, and results in a larger fracture height. In the three
cases, the fractures actually had some circumferential growth, but they were not able to
extend completely to the side of the minimum horizontal stress (Y-axis), even though there
is small resistance for fracture propagation.
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Figure 6.4: Developments of fracture geometry, fracture pressure and cement plastic strain
with different initial crack sizes. The pictures are front views of the one-
quarter model, with maximum horizontal stress in the X-axis direction and
minimum horizontal stress in the Y-axis direction. The circumferential
extents of the initial cracks for cases Q1 through Q4 are 30𝑜, 45𝑜, 60𝑜 and
90𝑜, respectively, as given in Table 5.5.
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6.3.3 Cement Interface Debonding with Different Cement and Formation
Properties
To investigate the influence of cement properties on the propagation of debonding
fracture due to excessive pressure buildup around the casing shoe, three types of cement
system with different stiffness, as given in Table 6.1, are considered. Type C3 corresponds
to “stiff-cement” with a high value of Young’s modulus and a high value of cohesive
strength. To the contrary, type C1 is “compliant-cement” with lower values of Young’s
modulus and cohesive strength. Properties of cement type C2 are in between those for stiff-
and compliant-cement. All the other input variables are according to values of the base
cases provided in Section 6.2.
Figure 6.5 shows the fracture growth and pressure distribution at the cement
interface for the three cases with different cement stiffness by the end of the second
simulation step. The debonding fracture with stiff cement has a larger growth in height than
the facture with soft cement. This is because stiffer cement restricts fracture opening in the
radial direction of the wellbore. Thus, with the same amount of fluid leakage into the casing
shoe, the stiffer cement shows a larger fracture height.
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Figure 6.5: Developments of fracture geometry and fracture pressure with different cement
properties. The pictures are front views of the one-quarter model, with
maximum horizontal stress in the X-axis direction and minimum horizontal
stress in the Y-axis direction. The properties for cement types C1 through C3
are given in Table 6.1.
Effects of formation properties on debonding fracture propagation were also
investigated. Similar to the analysis of cement properties, three types of formation system
with different stiffness as given in Table 6.2 are studied. Formation type F3 corresponds to
“stiff-formation” with high values of Young’s modulus and cohesive strength, and
formation type F1 is “compliant- formation” with lower Young’s modulus and cohesive
strength. Properties of formation type F2 is in between of them. All the other input variables
are based on the values of base cases provided in Section 6.2. Figure 6.6 illustrates the
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simulation results of the three cases. It shows a strong dependence of the development of
fracture height on the stiffness of the formation rock. The higher stiffness (i.e. the higher
the Young’s modulus and the higher the cohesive strength) of the formation, the larger the
fracture height that will develop. This result can be explained with the same statement that
was made above for the influence of cement stiffness on fracture growth.
Figure 6.6: Development of fracture geometry and fracture pressure with different
formation properties. The pictures are front views of the one-quarter model,
with maximum horizontal stress in the X-axis direction and minimum
horizontal stress in the Y-axis direction. The properties for formation types
F1 through F3 are given in Table 2.
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6.4 SUMMARY
A three-dimensional finite-element model is developed to simulate the cement
debonding due to the propagation of a fluid-driven fracture along the cement interface. The
model successfully takes into account the main elements for well integrity analysis, such
as interaction between different materials, poromechanical nature of the cement, plastic
deformation of cement and formation, three dimensionality and anisotropy of in-situ
stresses, propagation of debonding fracture, and flow of fracturing fluid. The model was
used to quantify fracture geometry and fracture pressure distribution for various conditions
with different in-situ stresses, pre-exiting cracks at the casing shoe, and cement and
formation properties. The results show that non-uniform debonding fractures occur under
these conditions. With anisotropic horizontal stress, the debonding fracture has a smaller
width in the direction of the maximum horizontal stress and a larger width in the direction
of the minimum horizontal stress. With initial cracks in the cement interface, debonding
fractures tend to develop vertically along the axis direction of a vertical well, rather than
circumferentially around the well. The results also demonstrate that the debonding fracture
propagation is highly influenced by the stiffness of cement and formation. The proposed
model provides a useful tool for simulating the debonding of cement interface caused by
leakage and pressure buildup around the casing shoe. It is useful for evaluating the risk of
cement sheath failure and fluid leakage from cement interface during drilling, pressure
tests, perforation, hydraulic fracturing, and any kinds of fluid or gas injection operations
during the production phase.
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CHAPTER 7: Role of Field Injectivity Tests on Combating Lost
Circulation4
Fracture parameters such as fracture initiation and propagation pressure and in-situ
stresses such as minimum horizontal stress and overburden, are critical inputs for
petroleum exploration design. They are particularly important for lost circulation
evaluations and wellbore strengthening applications. Field injectivity tests during drilling,
including leak-off test (LOT), extended leak-off test (XLOT), and pump-in and flow-back
test (PIFB), are promising and highly reliable methods for determining fracture parameters
and/or underground stress information. Moreover, such tests are extremely helpful for
understanding the fracture growth mechanism from the wellbore wall to the far field region,
which is equally important for study of lost circulation and wellbore strengthening. This
chapter highlights the importance of field injectivity tests for understanding the
fundamentals of lost circulation and wellbore strengthening, with a review of different
kinds of field tests and a discussion of their advantages and limitations. A coupled fluid
flow and geomechanics injectivity test model is also developed which can capture the key
elements of injectivity tests known from field observation and aid the interpretation and
design of field tests.
4 Parts of this chapter has been published in the following conference papers which were supervised by K.
E. Gray:
Feng Y., Jones J. F., Gray K. E. "Pump-in and Flow-back Tests for Determination of Fracture Parameters
and In-situ Stresses" presented at the 2015AADE National Technical Conference and Exhibition, San
Antonio, Texas, April 8-9, 2015.
Feng Y., Gray K. E. “Signatures and Interpretations of Pump-in and Flow-back Tests in High
Permeability and Low Permeability Formations " presented at the 5th GEOProc International Conference
on Coupled Thermo-Hydro-Mechanical-Chemical (THMC) Processes in Geosystems, Salt Lake City,
Utah, February 25-27, 2015.
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7.1 INTRODUCTION
For drilling wells safely and efficiently, precise determination of minimum
horizontal stress and fracture parameters (e.g. fracture initiation and propagation pressure)
is critical for the drilling industry. These data are required for evaluating wellbore stability,
determining drilling mud weights, predicting wellbore breathing and lost circulation,
designing wellbore strengthening treatments, and selecting casing shoe depths.
Indirect methods from logs and geomechanical modeling are often used to predict
fracture gradient and minimum horizontal stress (Zoback, 2010). However, where a high
degree of precision is required, a direct field injectivity test measurement is usually needed
to calibrate the predicted results. Field injectivity tests performed in drilling operation may
include formation integrity tests (FITs), leak-off tests (LOTs), extended leak-off tests
(XLOTs), and pump-in and flow-back tests (PIFBs). But not all these tests can provide
reliable stress information, either due to insufficient injection time/volume (resulting in
insufficient fracture length for stress evaluation), or due to a number of factors distorting
test signatures and leading to interpretation difficulties and uncertainties.
Misinterpretation of minimum horizontal stress and fracture parameters may cause
a number of drilling problems, while increasing non-productive time and well costs. If
fracture pressure is overestimated, unexpected lost circulation and wellbore breathing can
occur. Additional unplanned casing strings required to mitigate these problems, may in
extreme cases jeopardize the success of the well. Conversely, if fracture pressure is
underestimated, planned well costs will be unrealistically high. As a result, other projects
may suffer from a lack of funding or the well may not even be drilled (Postler, 1997).
This chapter first reviews the different kinds of field injectivity tests used to
evaluate minimum horizontal stress and fracture parameters, with a description of their
advantages and limitations. Secondly, test signatures and factors that may lead to
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difficulties to injectivity test interpretation are discussed. Thirdly, a phase by phase
comparison and interpretation of two pump-in and flow-back test examples in different
formations is performed. The fourth objective of this chapter is to develop an injectivity
test simulation framework that can capture the key elements of a field injectivity test.
Finally, due to the similar fracture behaviors in an injectivity test and lost circulation, the
importance of the field tests for understanding the fundamentals of lost circulation and
wellbore strengthening is highlighted.
7.2 A REVIEW OF FILED INJECTIVITY TESTS
With the increasing importance of geomechanics in the petroleum industry,
accurate determination of fracture parameters (e.g. fracture pressure/gradient) and in-situ
stress are increasingly appreciated by petroleum engineers. Therefore, more and more field
hydraulic fracturing tests are designed and performed to obtain field stress information. In
drilling and completions, lost circulation are strongly influenced by fracture pressure and
in-situ stresses. Field injectivity tests, including FITs, LOTs, XLOTs, and PIFBs, are
designed to determine fracture pressure and minimum horizontal stress during the drilling
process. These tests are typically performed after a casing string has been set, after drilling
10 to 20 feet of new formation (van Oort and Vargo, 2008).
7.2.1 Formation Integrity Test (FIT)
A FIT is a test to confirm the cement and formation integrity near the casing shoe,
and to ensure the fracture gradient at the shoe is sufficient to withstand any expected or
potential loads while drilling the subsequent hole section. In a FIT, the bottom-hole
pressure is gradually increased to a pre-determined value, which is lower than predicted
fracture initiation pressure. Typically no fracturing occurs during a FIT, and the wellbore
remains intact. The pressure-time curve remains as a straight line during the test, while the
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slope of the line indicates the stiffness/compliance of the wellbore systems. Accurate
fracture parameters or in-situ stress information cannot be obtained from a FIT.
7.2.2 Leak-off Test (LOT)
A LOT, as schematically shown in Figure 7.1, provides an estimate of fracture
gradient at the casing shoe. This information is used for mud weight design and lost
circulation prevention in the subsequent hole section. It is often assumed that the lowest
fracture gradient in a given wellbore will exist at the casing shoe, however for a variety of
reasons this is not always the case. During the LOT, the well is shut-in and drilling fluid is
gradually pumped into the wellbore at a constant small rate, until a noticeable inflection
point is observed on the pressure-time curve. The wellbore pressure at the inflection point
is known as leak-off pressure, indicating that a fracture has been created. Leak-off pressure
is commonly assumed equal to fracture initiation pressure, which is used to determine the
upper limit of wellbore fluid pressure.
However, it is the author’s contention that leak-off pressure is not necessarily equal
to fracture initiation pressure. It is usually somewhat higher than fracture initiation
pressure, especially when “dirty” fluid (i.e. drilling mud with high solids content) is used
for the test. This idea has been mentioned in Chapter 2 and will be further discussed latter
in this chapter.
It is worth noting that a leak-off test rarely provides far-field stress information,
since the fracture created in a leak-off test remains very short (within several radii from the
wellbore) due to minimal injection volume. This short fracture is still under the influence
of the near wellbore stress concentration. Figure 7.2 shows the hoop stress concentration
in the near wellbore region, for a vertical well in a formation with uniform far-field
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horizontal stresses. To correctly measure the far-field minimum horizontal stress, a fracture
must extend through the near wellbore stress concentration region.
Figure 7.1: Schematic illustration of pressure-time/volume plot in a leak-off test.
Figure 7.2: Near wellbore hoop stress concentration with uniform far-field stresses
𝑆ℎ𝑚𝑖𝑛 = 𝑆𝐻𝑚𝑎𝑥 (blue color = more compression, red color = less
compression).
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7.2.3 Mini-frac Test (MFT)
A mini-frac test (MFT) is a small scale fracturing test commonly performed before
a conventional hydraulic fracturing stimulation operation. Fracture gradient and minimum
horizontal stress can be determined from a MFT, and then used for hydraulic fracturing
design. However, MFTs are usually performed in reservoir formations, and very commonly
in cased and perforated well intervals. So, the prediction results from MFTs cannot
represent the situation for the entire well section, most of which is in non-reservoir
formations, where most lost circulation events occur.
7.2.4 Extended Leak-off Test (XLOT)
A fracture generated during an XLOT will typically have sufficient length to pass
through the near wellbore stress concentration region. Therefore, this test may provide
sufficient data for estimating far-field stress and fracture information. In an XLOT, fluid
injection continues until a relatively steady fracture propagation pressure is reached,
followed by a shut-in phase as shown in Figure 7.3. In addition to leak-off pressure, several
other fracture parameters can be estimated from an XLOT, including formation breakdown
pressure, fracture propagation pressure, instantaneous shut-in pressure and fracture closure
pressure.
Formation breakdown pressure is the divide separating the stable and unstable
fracture propagation stages. During the time between leak-off and formation breakdown,
the fracture experiences stable propagation. The fracture growth during this period is very
slow, with a volume increase rate less than the pumping rate. Therefore, the wellbore
pressure continues to rise prior to formation breakdown, which is the upper pressure limit
for stable fracture growth. The volume increase during this stage is primarily due to the
growth of fracture width rather than length.
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Subsequently, during the time between formation breakdown and fracture
propagation, the fracture extends unstably with a rapid increase in fracture length. The
wellbore experiences a sudden pressure drop immediately after formation breakdown,
since the fracture volume expands at a rate much greater than the pumping rate. With
continued injection, the pressure ultimately levels off to fracture propagation pressure,
indicating the fracture growth rate is roughly equal to the pumping rate. However, with
continued pumping and an increase in fracture length, the fracture propagation pressure
will gradually decrease in a wave-like pattern.
Instantaneous shut-in pressure (ISIP) is the pressure observed immediately after
pumping is stopped. Once pumping ceases, the additional pressure required to overcome
flowing friction along the fracture and tubing (if pressure is recorded at the surface) and
the fracture tip resistance, immediately drop to zero. Therefore, instantaneous shut-in
pressure is always somewhat lower than fracture propagation pressure.
Fracture closure pressure is frequently estimated from shut-in data using
interpretation methods developed for mini-fracturing tests. Fracture closure pressure is
commonly taken as the best estimate of minimum horizontal stress, based on the
assumption that the wellbore pressure is equal to minimum horizontal stress as the
mechanical fracture starts to close. It is worth noting that mini-fracturing test interpretation
methods were initially developed for diagnostic fracture injection tests in permeable
reservoirs with clean fluids. However, XLOTs are usually performed in tight shale with
mud. Therefore, direct use of these methods for XLOT interpretation may give inaccurate
results. In addition, fracture closure during the shut-in phase is the result of fluid leak-off
into the formation, which is highly dependent on permeability and threshold capillary entry
pressure. In low-permeability formations like tight shale, leak-off may be too slow for the
fracture to close within a reasonable amount of time. In this case, minimum horizontal
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stress cannot be properly evaluated using the aforementioned methods. Alternatively, an
XLOT with a flow-back phase may provide a better solution. This test is referred to as
pump-in and flow-back test (PIFB) in this chapter.
Figure 7.3: Typical XLOT plot (Modified after Gaarenstroom et al., 1993).
7.2.5 Pump-in and Flow-back Test (PIFB)
A pump-in and flow-back (PIFB) test is a standard XLOT followed by a flow-back
phase with either a constant choke or constant flow-back rate at the surface. Both XLOTs
and PIFB tests can include multiple cycles for confirming test results. Since fluids flow
back directly to the surface, fracture closure in a PIFB test is almost always assured and
not dependent on fluid leak-off into the formation. These tests provide a superior method
of measuring fracture closure pressure in low permeability formations, especially when
mud is used as the injection fluid. In a PIFB test, wellbore pressure and flow-back
time/volume are recorded. A plot of pressure versus flow-back time/volume will show an
inflection point, which indicates a change in system stiffness/compliance (see Figure 7.4).
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This inflection point is commonly interpreted as fracture closure pressure, equal to
minimum horizontal stress.
Figure 7.4: An example of pressure-volume plot in a pump-in and flow-back test
(Modified after Gederaas and Raaen, 2009).
In addition to minimum horizontal stress, PIFB tests also provide leak-off pressure
(fracture initiation pressure), formation breakdown pressure, fracture propagation pressure,
instantaneous shut-in pressure, and fracture reopening pressure. These values (especially
fracture initiation and propagation pressure) are important considerations for mud density
selection, casing program design and lost circulation prevention, particularly in wells with
narrow drilling margins.
With clean injection fluid, fracture initiation and propagation pressures are
primarily dominated by in-situ stresses and rock strength. Conversely, when drilling mud
is used as the injection fluid, these parameters are also highly dependent on mud properties
(e.g., mud type and solid particle size and concentration), petro-physical properties of the
rock other than strength (e.g., lithology, permeability, and wettability), and interactions
between the drilling mud and formation rock (e.g., fluid leak-off and filter cake
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development on the wellbore wall and fracture faces, and capillary pressure effects). A
detailed understanding of these and other factors is required for a proper interpretation of
PIFB tests, and understanding fracture growth behavior and lost circulation fundamentals.
7.3 TEST SIGNATURES
7.3.1 Pressure-time Signature
The plot of the pressure-time response of a typical PIFB test is shown in Figure 7.5.
With fluid pump-in, wellbore pressure first increases linearly to the LOP point. At the LOP
point, the straight line deviates to the right, and then continues to rise with a smaller slope
until it reaches the FBP. Immediately after FBP, the wellbore pressure experiences a
significant drop. Then, with continuous pumping, the pressure declines to the relative stable
FPP. At the beginning of the shut-in phase after stopping the pump, the wellbore pressure
has another sudden drop from FPP to ISIP. Then, the pressure decline becomes slower and
slower. Finally, a flow-back phase follows the shut-in phase. During the flow-back phase,
wellbore pressure first decreases with a relatively smaller rate, then with a relatively larger
rate.
The deviation at LOP reflects a significant change in the stiffness (or compliance)
of the pressure system. Several factors may influence this change, including the
compressibility of mud, casing, cement and formation rock, fluid penetration from
wellbore wall, and fluid leak-off into fractures. Among these factors, only the effect of
fluid leak-off into fractures is observably nonlinear (Fu, 2014). Therefore, when there is a
clear LOP, it is considered that a fracture must have been created.
After fracture creation, usually, the fracture first grows at a rate smaller than the
pumping rate. So, the wellbore pressure continues to rise until FBP. FBP divides the stable
and unstable fracture propagation stages. From LOP to FBP, the fracture grows stably and
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slowly. After FBP, the fracture experiences an unstable propagation with a fast increase in
fracture length and volume. The fracture volume expands at a rate higher than the pump
rate, so there is a dramatic pressure drop after FBP.
After formation breakdown, the pressure finally declines to the relatively constant
FPP. The fracture growth rate is roughly equal to the pump rate. The wellbore pressure
features a “saw-tooth” shape during fracture propagation in field tests. An example is
shown in Figure 7.6.
With a rapid pressure drop, wellbore pressure declines to ISIP after the pump is
stopped. This pressure drop is caused by the elimination of frictional pressure and pressure
required to overcome fracture tip resistance. ISIP is often referred to as minimum fracture
extension pressure or fracture gradient within the fracture stimulation and well completion
communities. Some researchers argue that ISIP is a reasonable estimate of minimum
horizontal stress (Alberty et al., 1999; Postler, 1997).
FCP is commonly recognized as the best estimate of minimum horizontal stress
(Fjar et al., 2008; Okland et al., 2002). FCP can be obtained either by using shut-in data
(plotting pressure vs. square root of time), or by using flow-back data (plotting pressure vs.
flow-back time or volume). In both methods, a change in the slope of the data line indicates
fracture closure. The pressure at this slope-change point is FCP. However, fracture closure
is not an instantaneous process (Fjar et al., 2008; Hayashi and Haimson, 1991; Raaen et
al., 2001; Raaen and Brudy, 2001). So, the slope change is not an instantaneous process
too. According to Hayashi and Haimson (1991) fracture closure is a two-stage process: in
the first stage fracture width reduces while the fracture length remains constant; in the
second stage fracture length decreases until the complete closure of the fracture. It is argued
that the minimum horizontal stress should be interpreted at the end of the first stage, rather
than at full fracture closure (Raaen et al., 2006, 2001; Raaen and Brudy, 2001).
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Figure 7.5: Pressure-time plot of a typical PIFB test.
Figure 7.6: “Saw-tooth” pressure response during fracture propagation (Reproduced
from Okland et al., 2002).
7.3.2 Pressure-volume Signature
The pressure-volume response of a typical PIFB test is shown in Figure 7.7. 𝑉𝑖𝑛 is
the total volume pumped into the well during the pump-in phase in Figure 7.5. 𝑉𝑤 is
volume pumped into the well before a fracture has been created. It actually represents the
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combined volume change due to mud compression, casing expansion, fluid penetration
through wellbore wall, and open hole expansion, as schematically shown in Figure 7.8.
After LOP, a fracture must have been created, but at first its volume grows at a rate
lower than pump rate, resulting in a continuously increasing pressure until formation
breakdown pressure (FBP). The fracture during this stable propagation period remains
small. 𝑉𝑓𝑠 is equal to fluid volume leaked off into the fracture plus any further volume
change due to mud compression, casing expansion, fluid penetration, and open hole
expansion, with wellbore pressure increase.
The wellbore pressure drops from FBP to FPP during the unstable fracture
propagation stage. Thus, in this period, no further mud compression or wellbore expansion
occurs. 𝑉𝑓𝑝 represents the injection volume to extend the fracture and fluid penetration
loss to the formation through wellbore and fracture surfaces. It is suggested that this volume
should be used to check the length of the fracture to evaluate whether it is sufficient to pass
the near-wellbore stress concentration region and reach the far-field area (Fjar et al., 2008).
From instantaneously shut-in pressure (ISIP) to final shut-in pressure (FSIP), there
is no volume change in the system, because the well is shut in. But the pressure has a
significant drop due to the removal of frictional pressure and pressure required to overcome
fracture tip resistance.
During the flow-back phase, 𝑉𝑜𝑢𝑡 is the total volume flow back to the surface. 𝑉𝑓𝑐
is roughly equal to the volume flow back from the fracture during facture closure. 𝑉𝑟𝑜𝑐𝑘
is the additional volume returned after fracture closure, due to inward fluid flow from the
formation to the wellbore, and wellbore shrinkage with pressure decrease. 𝑉𝑙𝑜𝑠𝑠 is the total
fluid volume lost into the formation.
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Figure 7.7: Pressure-volume plot of a typical PIFB test (Modified after Fjar et al., 2008).
Figure 7.8: Total volume pumped into the well before fracture creation (Modified after
Altun, 1999 and Fu, 2014).
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7.4 TEST INTERPRETATION
7.4.1 FIP, LOP and FBP
In an idealized situation with perfectly impermeable rock, clean fluid and relatively
low horizontal stress anisotropy, FIP, LOP, and FBP of a vertical wellbore should be
identical, as shown in Figure 7.9. They all represent the pressure required to initiate a
fracture on wellbore wall.
Figure 7.9: FIP=LOP=FBP in idealized condition.
However, FIP can be much lower in permeable formations than in impermeable
formations as discussed in Section 2.3. This is because local pore pressure near the
wellbore can experience an increase in a permeable formation due to wellbore fluid
penetration. This elevated pore pressure opposes compression stress, thereby leads to a
decreased compression (or increased tension) in the vicinity of the wellbore, resulting in a
relatively low FIP. Conversely, in impermeable rock, there is no fluid penetration, hence,
no pore pressure increase near the wellbore, resulting in a relatively higher FIP.
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When drilling mud is used as the injection fluid, because it usually has high solids
content, a mudcake commonly develops on the wellbore wall in a permeable formation.
With extremely low permeability, mudcake can effectively isolate wellbore fluid and
formation fluid, and inhibit pore pressure increase near the wellbore as analyzed in Chapter
4. Therefore, it can help maintain a relatively high FIP for a permeable wellbore.
LOP is the clear point where the pressure vs. injection time/volume plot begins to
deviate from linearity, indicating a significant change in the stiffness or compliance of the
system. LOP is commonly assumed equal to FIP by many drilling engineers. But this
assumption is only correct when clean injection fluid without solids is used. For drilling
mud with high solids content, LOP is not necessary identical to FIP.
With clean fluid, when a micro-fracture is created on the wellbore wall, the fluid
can easily enter the fracture. Fluid pressure can immediately act on fracture surfaces and
fracture tip to propagate the fracture, leading to a noticeable LOP on the pressure response.
Conversely, when drilling mud is used as injection fluid, the mud solids can quickly seal a
newly created micro-fracture by forming a filter cake as mentioned in Section 2.3. The
filter cake can arrest further fluid flow into the fracture and isolate the fracture tip from
wellbore pressure. Due to sealing effect of the filter cake, micro-fracture initiation (i.e. FIP)
is usually not detectable in the pressure response in lab tests even with a precise gauge, let
alone in field tests, as discussed by Guo et al. (2014) based on their laboratory experiments.
They have also shown that a fracture can grow to a significant size without a detectable
LOP. With continuous injection, a LOP may be observed, but it can be much higher than
the actual FIP, at which a fracture is initially created.
The above analysis implies that it may not be possible to precisely measure FIP in
permeable rock with drilling mud as the injection fluid. An inflection point may be
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undetectable at FIP; or if detected, it may indicate a higher filter cake breakdown pressure
rather than FIP.
Similarly, FBP can also be significantly increased by filter cake development in the
fracture. This phenomenon is more obvious in high permeability formations than in low
permeability formations, since filter cake is easier to develop in the former case. The
difference between FIP (or LOP) and FBP is relatively large in permeable formations,
compared with less permeable formations. Figure 7.10 shows a more realistic case where
FIP, LOP and FBP are not identical in permeable formations when “dirty” drilling mud is
injected.
Figure 7.10: FIP, LOP and FBP are not identical with drilling mud as injection fluid.
7.4.2 FPP
For drilling engineering, FPP is critical for wellbore stability evaluation, lost
circulation prevention, and casing design, especially for drilling in challenging areas with
a narrow drilling mud weight window. In an idealized case, with clean injection fluid and
impermeable formations, FPP is primarily dominated by minimum horizontal stress, and it
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can be a reliable estimate of the minimum horizontal stress. However, when drilling mud
is used and the formation is permeable, fluid leak-off and filter cake development in the
fracture can significantly influence FPP.
During fracture propagation, FPP is the total pressure needed to overcome 1) the
minimum horizontal stress for keeping the fracture open, 2) frictional losses for fluid flow,
3) fluid loss to the formation due to fluid penetration through fracture surfaces, and 4)
fracture tip resistance for further fracture growth. At least the latter three components can
be affected by fluid leak-off and/or filter cake development during fracture propagation.
During pumping, solids in the mud are transported with fluid flow into the fracture.
While mud filtrate leaks off into the formation, the solid particles in the mud are left in the
fracture. This will result in a higher solid density and fluid viscosity in the fracture, which
will significantly increase the frictional losses when the fracture has a large length,
therefore, lead to a higher FPP.
With fluid leak-off from the fracture surfaces to the formation rock, the fluid
volume (energy) inside the facture acting to extend the fracture is reduced. Therefore, for
creating further fracture volume, a higher pressure (energy) is needed to make up this
energy loss due to fluid leak-off. This phenomenon also increases FPP for a permeable
formation.
Moreover, with fluid leak-off in permeable formations, filter cake can easily build
up at the fracture surfaces and fracture tip. It can effectively isolate the fracture tip from
wellbore pressure, thereby significantly reducing pressure acting to propagate the fracture.
Additional pressure is required to break the filter cake for further fracture propagation,
which leads to a higher FPP. Conversely, in impermeable formations, no effective mud
barrier can build up in the fracture because there is no filtrate leak-off. This means there is
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full pressure communication between wellbore and fracture tip. The fracture can extend at
a relatively lower pressure.
The performance of filter cake is critical for lost circulation prevention and
wellbore strengthening treatment. Since a quality filter cake can only form in permeable
formations, it is generally believed that wellbore strengthening treatments work better in
permeable formations (e.g. sand and sandstone). Conversely, in relatively impermeable
rocks (e.g. shale), wellbore strengthening is not likely to be successful. When permeability
is low, the filtrate leak-off rate is too low to allow mud solids to aggregate to form a quality
filter cake or bridge. Low leak-off rates also mean very limited fracture pressure/energy is
released into the formation. Therefore, pressure is trapped inside the fracture, facilitating
fracture growth and lost circulation. Morita et al., (1990) and van Oort et al. (2011) also
argued that an effective filter cake is more likely to build up in water based mud (WBM)
than in oil based mud (OBM). That’s one reason why lost circulation is more likely to occur
when drilling with OBM.
7.4.3 Estimation of Minimum Horizontal Stress
It is widely accepted that fracture closure pressure (FCP) is the best estimate for
minimum horizontal stress (Fjar et al., 2008; Fu, 2014; Gederaas and Raaen, 2009; Raaen
et al., 2001; Raaen and Brudy, 2001; Ziegler and Jones, 2014). In a PIFB test, FCP can be
evaluated either from shut-in data or from flow-back data. When using the shut-in data, the
leak-off rate from fracture to formation should be sufficiently high for the fracture to close
in a reasonable time. So the prediction results depend highly on rock permeability. For the
latter method using flow-back data, fracture closure is almost guaranteed and less
dependent on fluid leak-off (permeability), because fluid can directly flow back to the
surface, rather than to the formation.
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For formations with high permeability and tested with a low-viscosity and low-
solids-content fluid, shut-in data can give a reasonable estimate of minimum horizontal
stress, because the fracture can close in a relative short time after the pump is stopped, due
to rapid leak-off. A highly sloped pressure-time curve indicating a fast pressure drop during
shut-in is expected (see Figure 7.11). Conversely, in formations with low permeability,
such as shale, the fracture usually does not close in a reasonable time due to the low leak-
off rate; therefore, shut-in data are not sufficient to give a good estimate of minimum
horizontal stress. A relatively flat pressure-time response is usually observed for this case
(see Figure 7.11).
However, even if the formation has sufficient permeability for fluid leak-off,
fracture closure can be significantly inhibited by a tight filter cake on the fracture surfaces.
In other words, the minimum horizontal stress measurement is also highly related to filter
cake quality in the fracture. With a tight mudcake, the leak-off may be too slow to close
the fracture. This implies that FCP in the shut-in curve may not be properly defined, similar
to the impermeable case in Figure 7.11.
It should also be noted that the methods for estimating FCP using shut-in data are
based on interpretation methods developed for MFTs. These methods, such as pressure vs.
square root of time plotting technique, were initially developed for pressure transition
analysis in permeable reservoirs with relatively clean fracturing fluid. However, XLOTs
or PIFB tests are usually performed in tight shale with “dirty” mud. So, the direct use of
these methods for shut-in data interpretation in low permeability formation with drilling
mud may lead to inaccurate results.
For a better estimation of minimum horizontal stress, especially for low permeable
formations or permeable formations with well-developed filter cake, a flow-back test is
preferred.
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In a flow-back test, a plot of pressure versus flow-back time will show an inflection
point, as shown in Figure 7.5, which indicates system stiffness/compliance change due to
fracture closure. The slope before this point reflects the stiffness/compliance of system
with an open fracture, while the slope behind this point is the stiffness/compliance of
system with a closed fracture. The pressure at this inflection point is FCP, which is
commonly recognized as a good estimate of minimum horizontal stress.
Raaen et al. (2001) and Gederaas and Raaen (2009) also suggested recording the
flow-back volume data. A plot of pressure versus flow-back volume helps determine the
FCP, as shown in Figure 7.12. They argued that a flow-back test gives more precise
estimate of minimum horizontal stress than traditional methods using shut-in data, which
usually overestimate the minimum horizontal stress.
Figure 7.11: Permeable formation has a large pressure decline during shut-in due to
sufficient leak-off from fracture, while a relative flat pressure response is
usually observed in an impermeable formation.
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Figure 7.12: An example of predicting FCP using flow-back data (Reproduced from
Gederaas and Raaen, 2009).
7.5 FIELD EXAMPLES
Discussion in this section concerns two PIFB tests obtained from the literature. The
two tests were conducted in two neighboring wells, Well 11-2 and Well 10-7, in the North
Sea (Okland et al., 2002). The test in Well 11-2 was performed in a formation with apparent
higher permeability compared with the test formation in Well 10-7. The two tests were
performed at 4097 ft and 4284 ft true vertical depth (TVD), respectively. Test in Well 11-
2 has two cycles, with pump-in and shut-in phases in the first cycle and pump-in and flow-
back phases in the second cycle (Figure 7.13). Test in Well 10-7 also has two cycles, but
there is an additional flow-back phase in the first cycle (Figure 7.14). From Figures 7.13
and 7.14, it can be seen that signatures of the two tests in high and low permeability
formations are very different, which are discussed and compared in this section.
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Figure 7.13: Test in Well 11-2 in formation with relatively high permeability
(Reproduced from Okland et al., 2002).
Figure 7.14: Test in Well 10-7 in formation with relatively low permeability (Reproduced
from Okland et al., 2002).
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Pump-start to FBP. In the phase from pump-start to FBP, it is obvious that there
is a relatively clear LOP in Well 10-7 compared with Well 11-2. In Well 11-2 the
pressure builds up linearly to FBP without a detectable deviation. A possible
explanation for this is that a leak-off response may not be detectable due to the
continuous sealing effect of filter cake in relatively permeable formation (Well 11-
2), as discussed above. Another observation is that FBP in the high permeability
case is much lower than that in the low permeability case, even though the test
formation depths are not much different. This may be related to the permeability of
the formation. High permeability formations (e.g. sand or sandstone) usually cannot
transmit as much overburden to horizontal stress as low permeability formation
(e.g. shale) do. Therefore, the test formation in Well 11-2 may have a relatively
lower in-situ stress due to its relatively higher permeability, thereby resulting in a
low FBP.
Fracture propagation in first cycle. Comparing the two cases, it can be found
that, after formation breakdown, the pressure drop in the low permeability
formation is much faster than that in the high permeability case. This may be related
to a relative higher fluid leak-off rate on fracture faces in the more permeable
formation. This leak-off reduces the energy acting on creating new fracture volume,
thereby resulting in a slow pressure drop. Another possible reason is that mud solids
form a filter cake with fluid leak-off, which seals the fracture tip and inhibits
fracture propagation. Comparing FBP and FPP, the less permeable formation
experiences a larger pressure drop from FBP to FPP than that in the more permeable
formation. An explanation for this could be: compared with low permeability
formation, high permeability formation has a relatively lower FBP due to its lower
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in-situ stress, but a relatively higher FPP due to filter cake development; therefore,
it has a relatively smaller difference between FBP and FPP.
Shut-in and flow-back in first cycle. Pressure decreases quickly during shut-in in
Well 11-2 due to its high permeability, while remaining essentially constant in the
low permeability Well 10-7. This implies that the fracture probably closed during
shut-in for the high permeability Well 11-2, and a FCP (or minimum horizontal
stress) can be properly predicted using the shut-in data. However, the fracture is
less likely to close in the low permeability well due to the very limited fluid leak-
off. This further confirms that FCP (or minimum horizontal stress) cannot be
reasonably determined using shut-in data in relatively impermeable rocks; it also
confirms that the pressure analysis methods, developed for interpretation of MFTs
in high permeability reservoir where fractures can easily close, may not be suitable
for shut-in data analysis in low permeability formations, where most XLOTs and
PIFB tests are performed.
Fracture reopening and re-propagation in second cycle. In both cases, there is
a fracture reopening pressure (FRP) which is much lower than the FBP in the first
cycle. This is because a fracture is already created on the wellbore wall after the
first injection cycle, therefore no additional pressure is needed to overcome the rock
strength to initiate the fracture. In an idealized situation with clean injection fluid
and impermeable rock, the difference between FBP and FRP can be interpreted as
the tensile strength of the rock. In the low permeability test, FPP is almost equal to
FRP. But in the high permeability test, FPP is somewhat higher than FRP. This may
also be related to the sealing effect of filter cake in high permeability rock. After
fracture re-opening, the fracture has full conductivity in impermeable rock, and
fluid can flow freely from the wellbore to the fracture tip. However, filter cake in a
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permeable fracture may significantly restrain pressure communication between the
wellbore and the fracture tip, leading to an FPP higher than FRP.
Flow-back in second cycle. In both tests, the pressure decreases quickly and
nonlinearly during flow-back. An inflection point can be defined in both cases, as
shown in Figures 7.13 and 7.14. As discussed above, the inflection reflects the
system compliance/stiffness change before and after fracture closure. Thus, the
pressure at this point is interpreted as FCP and commonly used as a prediction of
minimum horizontal stress. In the low permeability case, FCP predicted from flow-
back data in the second cycle is consistent with that predicted from the first cycle.
This confirms the test results.
7.6 DEVELOPING A SIMULATION FRAMEWORK FOR INJECTIVITY TESTS
Knowledge of minimum horizontal stress (𝑆ℎ𝑚𝑖𝑛) is important in many aspects
during the life of oilfield development. It is a key factor for prediction of wellbore stability
and lost circulation in the drilling and completion stages (Raaen and Brudy, 2001). FITs
and LOTs are routinely performed at each casing shoe during drilling. However, the main
purpose of these tests is to verify quality of the cement at the casing shoe and strength of
the next borehole section to be drilled. As discussed above, these tests normally cannot
give precise stress information due to limited fracture distance (Raaen and Brudy, 2001;
Ziegler and Jones, 2014). To overcome these limitations, XLOTs and PIFB tests were
introduced for quantifying 𝑆ℎ𝑚𝑖𝑛 with a higher degree of precision. Since a sufficiently
long fracture can be created and sufficient data can be gathered from a PIFB test, it has
been considered a preferred method of obtaining 𝑆ℎ𝑚𝑖𝑛 (Kunze and Steiger, 1992; Raaen
and Brudy, 2001; Zoback and Haimson, 1982), especially for early stages of oilfield
development. In a PIFB test, fluid injection continues until a relatively
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steady fracture propagation pressure (FPP) is reached, followed by a shut-in
phase and a flowback phase. As mentioned in the above section, the basic idea to predict
𝑆ℎ𝑚𝑖𝑛 using PIFB data is that FCP is equal to 𝑆ℎ𝑚𝑖𝑛 (Raaen et al., 2001). However,
prediction of FCP may be difficult (Guo et al., 1993; Plahn et al., 1997), and this difficulty
contributes significantly to the uncertainty of stress measurements (Raaen et al., 2001).
FCP identification using PIFB data frequently borrows techniques from pressure
analysis of hydraulic fracturing which was initially developed for determining reservoir
limits from diagnostic fracture injection tests in permeable reservoir formation. A typical
method is identifying fracture closure by plotting bottom hole pressure versus the square
root of shut-in time, with the pressure value at the slope change point of the plot being
interpreted as FCP. The traditional methods usually require that the fracture can close with
sufficient fluid leak-off from the fracture to the formation during shut-in. If the formation
is sufficiently permeable, this requirement can usually be satisfied with a reasonably long
shut-in period, i.e. the fracture will eventually close.
However, PIFB tests are mostly performed in tight shale formations with very low
permeability (these formations are loosely referred as “impermeable” formations in the
following discussion), where the casing shoes are usually set. For PIFB tests in such
formations, the fluid leak-off during shut-in may be too slow for the fracture to close within
a reasonable amount of time. Therefore, 𝑆ℎ𝑚𝑖𝑛 cannot be properly evaluated using the
aforementioned methods based on the time development of pressure during shut-in. The
direct use of these traditional methods for interpreting 𝑆ℎ𝑚𝑖𝑛 in impermeable formations
may lead to significant errors, usually an overestimate of 𝑆ℎ𝑚𝑖𝑛 (Raaen et al., 2001).
Therefore, a flow-back phase can be included in the test to aid fracture closure for a better
stress measurement. Since fluid flows back directly to the surface, fracture closure in the
flowback phase is always assured and not dependent on fluid leak-off into the formation.
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Therefore, flowback tests provide a superior method of measuring 𝑆ℎ𝑚𝑖𝑛 in impermeable
formations.
A proper understanding of the differences between PIFB tests in permeable
formations and impermeable formations is important for test interpretation and
determination of 𝑆ℎ𝑚𝑖𝑛 . This section first illustrates fully coupled fluid flow and
geomechanics modeling of PIFB tests to investigate and compare the essential pressure
and fracture behaviors of the tests in permeable and impermeable formations. Then, the
reliability of several in-situ stress prediction methods using shut-in data and flowback data
is tested and discussed with the numerical simulation results.
7.6.1 Model Formulation
Pressure responses and fracture behaviors in PIFB tests are simulated using a
coupled fluid flow and geomechanics numerical model in this study based on the Finite
Element Method (FEM). A PIFB test system generally has three components: the well, the
fracture, and the formation. Figure 7.15 shows a typical configuration of the PIFB test
system. The following physical processes that happen in a PIFB test are included and
simulated simultaneously in the proposed numerical model.
Fluid flow in the drill pipe
Fracture propagation and fluid flow in the fracture during fluid injection
Formation rock deformation and pore fluid flow
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Figure 7.15: General PIFB test system with well, formation and fracture.
The first objective of this section is to develop a general finite-element model that
can take into account the main elements in a PIFB test process, such as fluid injection
through the drill pipe, fracture propagation, fluid flow in the fracture, pore fluid flow, and
rock deformation. The model should be able to capture the essential features of a PIFB test,
such as pressure vs time/volume signatures and fracture closure behaviors during the test.
The second objective is to apply this model to simulate and compare PIFB tests in
permeable and impermeable formations.
Fluid flow in the drilling pipe is modeled based on Bernoulli’s equation as
described in Section 3.2.1 using the pipe element technique in Abaqus®. Fracture
propagation and fluid flow in the fracture during the tests are modeled based on a cohesive
zone method using coupled pressure/deformation cohesive elements in Abaqus®. A
traction-separation constitutive law and a fluid flow constitutive law as described in
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Section 3.2.2 are used in the cohesive zone model. The deformation of the porous formation
and pore fluid flow are modeled based on poroelastic theory and Darcy’s law as described
in Section 3.2.3 using coupled pore pressure and deformation continuum elements in
Abaqus®.
7.6.1.1 Assumptions and Geometry of the Model
PIFB tests performed in vertical wellbores are simulated in this study. The
overburden stress is in the vertical direction and the other two principal stresses, i.e.
minimum horizontal stress and maximum horizontal stress, are in the horizontal plane. The
wellbore and surrounding formation are assumed to be in a plane-strain condition as shown
in Figure 7.16. Owing to symmetry, only one half of the wellbore and formation is used in
the model.
The wellbore radius is 0.1 m. The total size of the modeled domain is 60×20 m. In
formations with anisotropic horizontal stresses, it is well known that the fracture will
propagate in the direction perpendicular to Shmin during the test. Therefore a fracture path
in this direction is defined in the middle of the model, as shown in Figure 7.16. The
formation is modeled as an isotropic, poroelastic material, using coupled pore pressure and
deformation continuum finite elements; the fracture is modeled using layer pore pressure
cohesive elements; and the well is model with pipe elements. Since significant
stress/displacement gradients are expected in the wellbore vicinity, the mesh is refined
around the wellbore, as shown in Figure 7.16, to increase the computational accuracy. The
total numbers of elements and nodes of the model are 1320 and 1476, respectively.
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Figure 7.16: PIFB test model. Top: geometry and boundary conditions of the model;
Bottom: refined mesh around the wellbore.
7.6.1.2 Boundary Conditions of the Model
A symmetric boundary condition is defined on the left edge of the model. The
formation is assumed to be at a depth of 1000 m with a normal pore pressure of 10 MPa, a
minimum horizontal stress of 15 MPa and a maximum horizontal stress of 18 MPa. The
minimum and maximum horizontal stresses and undisturbed pore pressure (10 MPa) are
applied on the outer boundaries of the model, as shown in Figure 7.16. Initial pore pressure
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of 10 MPa is applied to the whole formation. Drilling fluid is injected into the well from
the top of the drill pipe with a rate of 1.6 gallon per minute. Note that since a plane-strain
assumption is made, this rate is the fluid volume injected to a formation of unit thickness.
It is also assumed that the well has been circulated enough to ensure it is filled up
with drilling fluid before the PIFB test. So an initial hydrostatic pressure profile is defined
along the well depth before injection starts. The pipe element at end of the drill pipe is tied
to the formation element on the wellbore wall to make sure the fluid pressure in the bottom
hole is equal to the pore pressure on the wellbore wall. Besides, a dynamic pressure that
equals the bottom-hole fluid pressure is imposed onto the inner wellbore wall to model the
pushing pressure applied by fluid column.
7.6.1.3 Material Properties Used in the Model
All of the material properties of permeable and impermeable formations used in the
simulations are the same, except the formation permeability. The permeability values used
for the permeable and impermeable formations are 5 mD and 0.05 mD respectively, which
are typical values corresponding to sandstones and tight shales. Table 7.1 summarizes all
the other material parameters used for the simulations.
Table 7.1: Material properties for the simulations.
Parameter Values Units
Young’s modulus 7000 MPa
Poisson’s ratio 0.2
Fluid density 1000 Kg/m3
Fluid viscosity 1 cp
Porosity 0.25
Tensile strength 1.8 MPa
Critical fracture energy 30 J/m2
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7.6.2 Results and Discussion
Using the coupled fluid flow and geomechanics model described above, PIFB tests
in permeable and impermeable formations are simulated and analyzed. Several methods
for predicting 𝑆ℎ𝑚𝑖𝑛 using pressure versus time or pressure versus volume data from PIFB
tests are analyzed based on the simulation results. The summary of the analyses is reported
in the following subsections.
7.6.2.1 Simulated Test Signatures
PIFB tests with two cycles are simulated in this study. Each cycle include three
phases: injection, shut-in, and flowback. Since the fracturing fluid leak-off rates in the
permeable and impermeable formations are significantly different, different time schedules
were used for PIFB tests in the two types of formations in order to obtain adequate data for
analyses.
For PIFB tests in the permeable formation: first circle includes 600s injection, 600s
shut-in, and 100s flowback; the second circle includes 800s injection, 600s shut-in,
and 200s flowback.
For PIFB tests in the impermeable formation: first circle includes 600s injection,
600s shut-in, and 400s flowback; the second circle includes 1000s injection, 800s
shut-in, and 900s flowback.
Bottom hole pressure (BHP) versus time curves for PIFB tests in the permeable and
impermeable formations are plotted in Figure 7.17. The plots capture some essential PIFB
test signatures learned from field tests.
The pressure increases linearly to formation breakdown pressure (FBP) in the early
time of the injection phase. Since the numerical model does not take into wellbore storage
effect into account, the pressure builds up very quickly. It can be observed that pressure
buildup in a permeable formation is slower than that in an impermeable formation. This is
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because the relatively higher leak-off rate on the permeable wellbore wall leads to faster
fluid loss to the formation, hence slower pressure buildup in the wellbore.
After formation breakdown, the pressure declines to a relatively constant FPP.
Values of FPP in permeable and impermeable formations are 2.5 MPa and 1.7 MPa higher
than 𝑆ℎ𝑚𝑖𝑛, respectively. These results are in qualitative agreement with the numerical
studies by Lavrov et al. (2015) where FPP was reported about 2-2.7 MPa above 𝑆ℎ𝑚𝑖𝑛,
and the field test observations reported by Raaen et al. (2006) and Okland et al. (2002)
where FPP is around 1-1.5 MPa above 𝑆ℎ𝑚𝑖𝑛. By comparing the two cases, it can be seen
that a permeable formation has higher FPP compared to an impermeable formation with
the same in-situ stress condition. This can be explained as, due to the increased total stress
around the fracture with fluid leak-off, the permeable formation requires a higher BHP to
maintain a sufficient net fracture pressure for fracture propagation.
Figure 7.17 also shows that the BHP features a “saw-tooth” shape during fracture
propagation. This phenomenon again agrees well with field test observations by (Raaen et
al. (2006) and theoretical analysis by Feng et al. (2015b). This feature can be explained
from a fracture mechanics point of view. The fracture will have a growth in length when
the stored fracture energy reaches critical fracture energy. However, immediately after
length growth, the stored fracture energy will release to a value lower than the critical
fracture energy, so the fracture will stop propagating until enough fracture energy
(pressure) is achieved again with continued injection. The fracture growth and suspension
process will repeat during fracture propagation, resulting in a “saw-tooth” pressure pattern.
Another interesting observation is that the “saw-tooth” pressure fluctuation is more severe
in the permeable formation than in the impermeable formation. This again can be
interpreted by the higher fluid leak-off in the permeable formation, which results in an
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increased total stress in the surrounding formation acting on closing the fracture, and hence
a larger pressure buildup for fracture propagation.
During shut-in phase, the BHP in a permeable formation has a remarkable decline,
to a value much lower than the 𝑆ℎ𝑚𝑖𝑛 and approaching the formation pore pressure of 10
MPa; while in the impermeable test, the BHP only exhibits a very small decline and it is
always significantly higher than 𝑆ℎ𝑚𝑖𝑛. It is well known that the pressure drop during shut-
in phase is solely dependent on fluid leak-off. Since the impermeable formation has very
slow leak-off, the BHP keeps nearly constant during the entire shut-in phase. However,
during the flowback phase, BHP drops significantly below 𝑆ℎ𝑚𝑖𝑛 , because fluid flows
back directly to the surface and pressure drop is not dependent on fluid leak-off anymore.
In the shut-in and flowback phases, changes of slope are observed in the pressure vs time
curves, which are commonly used to predict 𝑆ℎ𝑚𝑖𝑛. Methods using shut-in and flowback
data to predict 𝑆ℎ𝑚𝑖𝑛 for permeable and impermeable formations are discussed further in
the following sections.
During the second test cycles, shut-in and flowback phases have very similar
pressure behaviors with those in the first cycles for both formation types. However, no FBP
is observed in the early injection time. Instead, an inflection point very close to 𝑆ℎ𝑚𝑖𝑛 is
shown in both cases. The pressure at this point can be interpreted as the pressure required
to reopen the fracture created in the first test cycle, and is commonly referred to as fracture
reopen pressure (FRP). The simulation results clearly indicate FRP is a good representative
of 𝑆ℎ𝑚𝑖𝑛.
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Figure 7.17: Bottom hole pressure versus time plots. Top: PIFB test in permeable
formation; Bottom: PIFB test in impermeable formation.
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7.6.2.2 Discussion on 𝑺𝒉𝒎𝒊𝒏 Prediction
𝑺𝒉𝒎𝒊𝒏 Prediction Based on Shut-in Data
As mentioned earlier, the basic idea to predict 𝑆ℎ𝑚𝑖𝑛 from PIFB tests is that the
FCP is equal to 𝑆ℎ𝑚𝑖𝑛.
Fracture closure, hence FCP, can be identified based on the time development of
pressure. A common method is using a plot of pressure versus square root of shut-in time.
A change in the slope of the plot is generally interpreted as the FCP. However, this method
is only valid when satisfying the following two prerequisites:
Linear fluid flow along the longitudinal direction of the fracture and Darcy’s Law
governed fluid leak-off into the formation;
Fracture can close due to fluid leak-off into the formation in a reasonable shut-in
time.
For permeable formations these requirements may be easily satisfied, while for
impermeable formations this method is not a desired tool for 𝑆ℎ𝑚𝑖𝑛 prediction due to its
negligible leak-off. Figure 7.18 shows the plots of BHP versus square root of shut-in time
for the two cases. It can be seen that there is a distinct inflection point on the permeable
test curve where the BHP equals to 15.4 MPa, which is a fairly reasonable estimate for
𝑆ℎ𝑚𝑖𝑛 (15 MPa). This pressure is therefore interpreted as FCP. This result is in agreement
with the previous studies on predicting 𝑆ℎ𝑚𝑖𝑛 using shut-in data in permeable formations
(Raaen et al., 2001). However, for the impermeable case, there is no clear inflection point
on the plot, and the pressure is considerably higher than 𝑆ℎ𝑚𝑖𝑛 through the entire shut-in
phase; therefore it is impossible to appropriately predict 𝑆ℎ𝑚𝑖𝑛 based on the time
development of pressure in such cases. This again agrees with previous study conclusions
by Raaen et al. (2006, 2001), Raaen and Brudy (2001) and Kunze and Steiger (1992).
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For the plot of the permeable test, the slope is relatively constant before FCP; and
then increases immediately after FCP; finally decreases to a nearly constant value again.
The instant slope increase can be interpreted as a result of system stiffness increase at
fracture closure. The slope decrease is caused by the reduced fluid leak-off due to the
reduced pressure difference between BHP and pore pressure at the later stage of shut-in.
Figure 7.18: Pressure versus square root of time plot during shut-in. Top: Permeable
formation; Bottom: Impermeable formation.
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𝑺𝒉𝒎𝒊𝒏 Prediction Based on Flowback Data
As mentioned above, PIFB tests are mostly performed in impermeable shale
formations. The fluid leak-off during shut-in may be too slow for the fracture to close
within a reasonable amount of time in such formations. In order to ensure fracture closure,
a flowback phase can be introduced in a PIFB test. In the numerical examples, a flowback
phase is added to each cycle of the test. Figure 7.19 shows the time development of BHP
during flowback in the impermeable formation. Clear infection points (FCPs) at the same
BHP are observed on both cycles. The FCP (15 MPa) accurately measures the minimum
horizontal stress 𝑆ℎ𝑚𝑖𝑛.
The slope of the curve at each side of the fracture closure point is a relatively
constant value, with a smaller slope before fracture closure and a larger slope after fracture
closure. This is consistent with existing observations from field tests (Raaen et al., 2001)
and numerical simulations (Lavrov et al., 2015). This slope change can be explained as:
the system has a relatively smaller stiffness with an open fracture before fracture closure,
resulting in a slower pressure decline with fluid flowback; while after fracture closure, the
system stiffness increases, leading to a faster pressure decline and a larger slope.
In field PIFB tests, it has been demonstrated that FCP can be precisely determined
using the pressure versus volume added to or flowback from the
wellbore/fracture/formation system (Gederaas and Raaen, 2009; Okland et al., 2002; Raaen
et al., 2006, 2001; Raaen and Brudy, 2001). This is also illustrated very well in the PIFB
test simulation in the impermeable formation. Figures 7.20 and 7.21 show the amount of
fracturing fluid stored in the system during shut-in and flowback phases of the impermeable
test and permeable test, respectively. Apparently, during shut-in, the added fluid volume
does not change, because there is no fluid leaving the system. The BHP exhibits a
significant reduction of 6 MPa in the permeable formation due to large leak-off, but only a
253
negligibly small decline of 0.3 MPa in the impermeable formation. It is evident from Figure
7.20 that the FCP (15 MPa) obtained from the pressure versus volume curves can give a
precise identification of the 𝑆ℎ𝑚𝑖𝑛. Again, before fracture closure a larger slope is observed
because of the relatively lower stiffness of the system with an open fracture; while after
fracture closure a smaller slope is observed due to the relatively larger stiffness of the
system with a closed fracture. Figure 7.21 shows that after the fracture has closed and the
pressure drops below 𝑆ℎ𝑚𝑖𝑛 in the shut-in phase in a permeable test, adding a flowback
phase does not provide any valuable information for stress determination, if not perturbing
the test interpretation. However, if the fracture is not closed due to limited shut-in time,
adding a flowback phase for tests in permeable formations may yield the same benefits as
for the impermeable formations. Therefore, whether for tests in permeable or impermeable
formations, it is important to properly schedule the shut-in and/or flowback periods to make
sure the fracture will close in the tests. The proposed model can be used to predict the time
required for the fracture to close in a shut-in test or in a flowback test based on the
formation properties and operational parameters.
254
Figure 7.19: BHP vs time during flowback phases in impermeable formation.
Figure 7.20: BHP versus added fluid volume in the system during shut-in and flowback
phases in the impermeable test.
255
Figure 7.21: BHP versus added fluid volume in the system during shut-in and flowback
phases in the permeable test.
7.7 LOST CIRCULATION AS A FUNCTION OF FORMATION LITHOLOGY
The fracture behavior in lost circulation and wellbore breathing is similar to that in
a field injectivity test. A fracture initiates at the wellbore wall, and is then propagated to
the far field by a fluid driven force. The main difference is that field injectivity tests are
generally conducted at a constant rate into a known formation and depth. On the other hand,
lost circulation generally occurs while circulating at a dynamic wellbore pressure. The
formation and depth where lost circulation is occurring is often unknown (at least initially)
especially in long sections of open hole. It is not always correct to assume lost circulation
is occurring near the previous casing shoe, where fracture gradient is presumed lowest.
Factors that affect field injectivity tests also influence lost circulation. From the
discussions above in this chapter and Chapter 2, permeability, capillary entry pressure, and
mudcake development on the wellbore wall and in the fracture can significantly affect
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fracture initiation and propagation pressures and therefore, lost circulation. These factors
can vary significantly for different formations and fluid types.
Accurate identification of formation types where lost circulation is more likely to
occur is extremely important for successful drilling operations (Ziegler and Jones, 2014).
In the following section, we will discuss the relative likelihood of lost circulation in salt,
clean shale, clean sandstone, and silty-shale. Fractured carbonates are intentionally omitted
from this discussion, since they are better handled as a separate topic.
Massive Salt. Due to its high plasticity, in-situ stresses in massive salt (halite) tend
to be uniform. Simple rock mechanics models show that under the same load condition (i.e.
overburden) uniform stresses lead to higher fracture initiation pressures, compared to
anisotropic stresses. Additionally, massive salt has extremely low permeability and is
therefore not penetrated by the drilling fluid. Lost circulation is rarely a problem in massive
salt, except under certain conditions where inclusions or sutures are present.
Clean Shale. Clean shale (typically high clay content, high Poisson’s ratio, low
Young’s modulus mud rock) also has very low permeability and relatively high plasticity
and fracture initiation pressure. Due to its small pore throat sizes, clean (usually water-wet)
shale has relatively large capillary entry pressures for immiscible oil or synthetic based
drilling fluids. The combination of low permeability and high capillary entry pressure
greatly inhibits fluid invasion into clean shale, preserving its relatively high fracture
initiation pressure. Compared to silty-shale, sandstone, siltstone and fractured carbonates,
lost circulation is usually much less of a problem in clean shale.
Sandstone. In theory, sandstone has relatively low fracture initiation and
propagation pressures compared to shale, silty-shale and salt (Ziegler and Jones, 2014).
However, due to its high permeability, large pore throat sizes and low capillary entry
pressures, a high quality filter cake can easily develop on the wellbore wall or inside a
257
fracture. This filter cake effectively maintains fracture initiation pressures and can
significantly increase fracture propagation pressure, if a fracture is created. Therefore, lost
circulation is usually manageable in sandstone if appropriate drilling fluids (including
LCM) are used. Pressure depleted sandstones present an additional challenge, since a
reduction in pore pressure will also reduce fracture initiation and propagation pressures.
Fracture initiation pressures may also be reduced depending on wellbore trajectory and the
relative magnitudes of principal in-situ stresses. The last statement is true for all formation
types, except where in-situ stresses are uniform or nearly uniform.
Silty-shale. Silty-shale is a term commonly used to describe formations with
properties between those of clean shale and sandstone. It therefore encompasses a relatively
wide range of formation types and is not a precise lithological description. Silty-shale is
commonly encountered in transition zones between clean shale and sandstone, but it can
occur anywhere in the wellbore. Many reservoir top seals in deep water are comprised of
shale, with silt contents ranging from 17 to 41 percent (Dawson, 2004). Therefore, much
analysis has been performed on these types of formations, with respect to mineralogy,
capillarity and reservoir fluid sealing capacity.
Silty-shale has intermediate permeability and capillary entry pressures, compared
with clean shale and clean sandstone. However, capillary entry pressures for water-wet
silty-shale are too high to be ignored, and likely hold the key to understanding the
differences in apparent fracturing behavior, when silty-shale is drilled with oil or synthetic
based fluid versus water based fluid.
When water-wet silty-shale is drilled with water based fluid, it generally behaves
much like permeable (albeit tight) sandstone, with an effective filter cake formed on the
wellbore wall or inside a fracture. Therefore, fracture initiation and propagation are no
more likely to occur than in sandstone.
258
However, when water-wet silty-shale is drilled with oil or synthetic based fluid,
capillary entry pressures are high enough to significantly inhibit leak off and filter cake
development, similar to clean shale but with lower fracture initiation and propagation
pressures. Therefore, without a protective filter cake, water-wet silty-shale drilled with oil
or synthetic based fluid, is the most likely formation type to undergo hydraulic fracture
initiation and propagation. Since permeability is higher than for clean shale, it may also be
possible that wellbore pressure can be communicated to both the wellbore wall and
potentially to the far-field, via natural fractures or interconnected pore space. These
conditions also mean that both preventive and remedial wellbore strengthening methods
are less likely to be successful.
Wellbore breathing, which is loosely described as fluid losses while circulating,
followed by fluid flow back when the pumps are stopped, is almost always associated with
oil or synthetic based fluids and water-wet silty-shale. Wellbore breathing is very similar
to a pump-in and flow-back test, where a fracture is initiated and propagated while
circulating. Since capillary entry pressures prevent fluid penetration and leak-off inside the
fracture, fluids lost while circulating return to the wellbore when the pumps are stopped
and the fracture closes.
Figure 7.22 shows logging data from two lost circulation and wellbore breathing
events, in the same wellbore, drilled with synthetic based drilling fluid. Both loss zones are
clearly identified by repeated resistivity measurements, while the gamma ray data clearly
indicates that both fluid loss events occurred in silty-shale, rather than clean shale or
sandstone.
259
Figure 7.22: Two lost circulation and wellbore breathing events occurred in silty shale
formations, rather than in clean shale or clean sand formations.
7.8 SUMMARY
Various types of field injectivity tests, for interpreting fracture parameters and
minimum in-situ stress, are reviewed in this chapter, including FITs, LOTs, XLOTs, and
PIFB tests. Effects of several key factors, such as formation permeability, fluid leak-off
and filter cake development, on field injectivity tests are discussed. It has been found that
FIP, LOP, FBP, FPP, and FCP are all related to formation permeability, fluid leak-off and
filter cake sealing. Ignoring their influences may lead to incorrect interpretation of field
injectivity tests, and consequent drilling problems, e.g. lost circulation, and unnecessary
cost. Disregarding fractured carbonates, lost circulation and wellbore breathing are highly
related to formation and fluid type and are most likely to occur in water-wet silty-shale
drilled with oil or synthetic based fluids.
260
A fully coupled fluid flow and geomechanics model for numerical simulation of
field injectivity tests is presented. The model successfully takes into account the main
elements in a field injectivity test, including fluid flow in the well, fracture propagation,
fluid low in the fracture, pore fluid flow and deformation of formation rock. The essential
signatures of field injectivity tests known from previously reported field tests can be
captured by the model. PIFB tests in permeable and impermeable formations are simulated
and compared using the model. It is demonstrated by the simulation results that an accurate
𝑆ℎ𝑚𝑖𝑛 may be determined in a permeable formation using traditional methods based on the
time development of pressure; while for impermeable formations where most of the field
tests are performed during drilling, a flowback test may be needed for a better prediction
of 𝑆ℎ𝑚𝑖𝑛 due to limited fluid leak-off. It is important to properly design the shut-in and/or
flowback schedules to ensure fracture closure in a field test. The model presented in this
chapter provides a useful tool for optimizing field test design to ensure that sufficient and
high-quality data are obtained for test interpretation.
261
CHAPTER 8: Conclusions and Future Work
This chapter first summarizes the major research work and the key conclusions of
this dissertation. Then, some recommendations are proposed for future work related to this
research.
262
8.1 CONCLUSIONS
I. The roles of fracture initiation and propagation pressures on lost circulation and
wellbore strengthening are first discussed in this dissertation. Factors that may affect
these two pressures are analyzed, which include micro-fractures on the wellbore wall,
in-situ stress anisotropy, pore pressure, fracture toughness, filter cake development,
fracture bridging/plugging, bridge location, fluid leak-off, rock permeability, pore size
of rock, mud type, mud solid concentration, and critical capillary pressure. The key
conclusions of this analysis included:
FIP of a wellbore with micro-fractures is controlled not only by pore pressure and
in-situ stresses, but also by fracture length and fracture toughness of the formation
rock. It can be much lower than that of a perfect wellbore.
Plugging a fracture can significantly increase its propagation pressure, especially
in formations with large differences between pore pressure 𝑃𝑝 and minimum
horizontal stress 𝑆ℎ𝑚𝑖𝑛.
Fluid leak-off through the fracture face hinders fracture growth by facilitating filter
cake development and reducing the fluid energy available to propagate the fracture.
Capillary entry pressure 𝑃𝑐𝑒 is an important and often neglected consideration for
lost circulation mitigation and wellbore strengthening. High capillary entry
pressures, associated with small pore openings and immiscible fluids, can restrict
fluid leak-off and filter-cake/plug development. Field observations indicate lost
circulation in fractured and silty shale formations occurs more frequently with
OBM/SBM than with WBM. Additionally, the observation that wellbore breathing
typically occurs in water-wet formations drilled with OBM/SBM may be elegantly
explained by capillary theory.
263
II. A finite-element framework to simulate lost circulation during drilling with circulation
of drilling fluid is developed. Circulation of drilling fluid in the “U-Tube” wellbore and
fracture propagation in the porous formation are coupled together to predict the
dynamic fluid loss and fracture geometry evolution in drilling process. The fluid flow
in the well is modeled based on the Bernoulli’s equation taking into account the viscos
loss. Fracture propagation and fluid flow in the fracture are modeled based on a pore
pressure cohesive zone method. The numerical model provides a unique new way to
model lost circulation in drilling when the boundary condition at the fracture mouth is
neither a constant flowrate nor a constant pressure, but rather a dynamic bottom-hole
pressure or ECD. Factors that can affect ECD and thus lost circulation, including mud
density, mud viscosity, pump rate and annulus clearance, are investigated using the
proposed model. The key conclusions of this work include:
The viscous pressure loss term due to fluid circulation in the annulus can lead to
significant ECD increase and fluid loss. Drilling mud with relatively low density
which does not cause lost circulation in static state without circulation may lead to
significant fluid loss after resuming mud circulation. So it is important to take into
account the dynamic circulation effect on lost circulation prediction and prevention.
In drilling operations, we are interested in preventing fractures from occurring by
controlling ECD, or plugging the fractures at the early time of their growth using
LCMs, so the capability of capturing the dynamic fluid loss and fracture geometry
development of the proposed framework can help us understand how to prevent
lost circulation, optimize mud rheology, and select LCMs.
III. An analytical solution and a numerical model are developed to investigate the role of
mudcake on preventive wellbore strengthening treatments based on plastering wellbore
with mudcake. The analytical solution derived based on steady-state fluid flow
264
assumption incorporates the effects of mudcake thickness, permeability and strength
on wellbore stress and fracture pressure. In order to describe time-dependent mudcake
effect on preventive wellbore strengthening. A finite-element framework based on
poroealstic theory is developed to investigate the transient effects of dynamic mudcake
thickness buildup and permeability reduction on the near-wellbore stress and pore
pressure, and thus the strengthening of wellbore. The key conclusions of this study
include:
Both mudcake thickness and permeability have great influence on wellbore stress
and fracture pressure. With the decrease of mudcake permeability and/or increase
of mudcake thickness, fracture pressure increases.
Mudcake strength has negligibly small effects on wellbore stress and fracture
pressure, and thus wellbore strengthening.
Taking into account the dynamic mudcake thickness buildup and permeability
reduction results in a time-dependent wellbore stress state between that without
considering mudcake and that assuming an impermeable mudcake.
The time-dependent mudcake model provides a useful tool to analyze the stress
evolution around wellbore with dynamic mudcake development for the design and
evaluation of preventive wellbore strengthening treatments based on plastering
wellbore surface with mudcake.
IV. An analytical solution and a finite-element model are proposed for modeling remedial
wellbore strengthening treatment based on plugging/bridging lost circulation fractures
using LCMs. The analytical model, based on linear elastic fracture mechanics theory,
provides a fast procedure to predict fracture pressure change before and after fracture
bridging. The numerical model, taking into account poromechanical effects, provides
a more accurate prediction of the distribution of local stress with remedial wellbore
265
strengthening operations. Sensitivity analyses are performed using both of the
analytical and numerical models to quantify the effects of rock properties, in-situ
stresses, bridge locations and fluid flow on remedial wellbore strengthening. Key
conclusions of this study include:
Fracture pressure can be significantly increased by bridging the small pre-existing
fractures emanating from the wellbore wall in remedial wellbore strengthening
operations.
The closer the bridging location to the fracture mouth, the better strengthening can
be achieved. This means for better application of wellbore strengthening
techniques, it is important to accurately predict fracture geometry, especially
fracture mouth opening, for selecting the best LCM size.
Remedial wellbore strengthening applications by bridging pre-existing fractures are
more effective for formations with small in-situ stress anisotropy than those with
large stress anisotropy.
Remedial wellbore strengthening applications are more effective for formations
with low pore pressure and in-situ stress ratio, such as pressure depleted reservoirs,
as compared to formations with large pore pressure and in-situ stress ratio, such as
deepwater high pressure formations.
After bridging a fracture, there is a compression increase area near the bridging
location, whereas the tensile stress near the fracture tip decreases. This means that
the fracture becomes more difficult to reopen and propagate, and lost circulation is
less likely to continue.
Fluid leak-off affects both hoop stress and fracture width distributions. It is
important to consider fluid leak-off on the wellbore wall and fracture faces in
266
predicting fracture geometry, optimizing LCM size distribution, and evaluating
potential hoop stress enhancement with wellbore strengthening.
For effectively strengthening the wellbore by preventing fluid communication
between the wellbore and fracture tip, a desirable LCM bridge should have low
enough permeability to ensure there is no fluid flow across the bridge and the
fracture portion behind the bridge can close due to fluid leak-off. The permeability
of the bridging plug is likely a more important parameter than its strength.
V. A three-dimensional finite-element framework is developed to exam the possibility of
fluid leakage through casing shoe and along the weak cement interface when there is
pressure buildup in the wellbore due to change of drilling or completion fluid,
conduction of injectivity tests, and etc. The model is used to quantify the length, width,
and circumferential coverage of the cement interface debonding fractures. The key
conclusions of this work include:
Non-uniform debonding fractures may occur under various conditions with
different in-situ stresses, pre-exiting cracks at the casing shoe, and cement and
formation properties
With anisotropic horizontal stress, the debonding fracture has a smaller width in the
direction of the maximum horizontal stress and a larger width in the direction of
the minimum horizontal stress.
With initial cracks in the cement interface, debonding fractures tend to develop
vertically along the axis direction of a vertical well, rather than circumferentially
around the well.
The debonding fracture propagation is highly influenced by the stiffness of cement
and formation.
267
The proposed model provides a useful tool for simulating the debonding of cement
interface caused by leakage and pressure buildup around the casing shoe. It is useful
for evaluating the risk of cement sheath failure and fluid leakage from cement
interface during drilling, pressure tests, perforation, hydraulic fracturing, and any
kinds of fluid or gas injection operations during the production phase.
VI. Finally, importance of field injectivity tests for understanding the fundamentals of lost
circulation and wellbore strengthening are highlighted, with a review of different kinds
of field tests and a discussion of their advantage and limitations. A coupled fluid flow
and geomechanics is also developed to simulate injectivity tests with pump-in, shut-in
and flowback phases. The key conclusions of this study include:
FIP, LOP, FBP, FPP, and FCP in an injectivity test are all related to formation
permeability, fluid leak-off and filter cake sealing. Ignoring their influences may
lead to incorrect interpretation of injectivity tests, and consequent drilling
problems, e.g. lost circulation, and unnecessary cost.
Accurate identification of formation types where lost circulation is more likely to
occur is extremely important for successful drilling operations. Disregarding
fractured carbonates, lost circulation and wellbore breathing are highly related to
formation and fluid type and are most likely to occur in water-wet silty-shale drilled
with oil or synthetic based fluids.
The injectivity test simulation model can capture the key elements of injectivity
tests known from field observation and aid the interpretation and design of field
tests.
It is demonstrated by the injectivity test simulation results that an accurate 𝑆ℎ𝑚𝑖𝑛
may be determined in a permeable formation using traditional methods based on
the time development of pressure; while for impermeable formations where most
268
of the injectivity tests are performed, a flowback test may be needed for a better
prediction of 𝑆ℎ𝑚𝑖𝑛 due to limited fluid leak-off. It is important to properly design
the shut-in and/or flowback schedules to ensure fracture closure in a field test.
8.2 FUTURE WORK
This dissertation made an attempt to perform a systematic study on lost circulation
and wellbore strengthening. Several analytical and numerical models were developed to
model dynamic fluid loss while drilling, preventive wellbore strengthening based on
plastering wellbore with mudcake, and remedial wellbore strengthening based on
bridging/plugging lost circulation fractures. Some recommendations for future research
related to this dissertation are described as following:
For modeling lost circulation while drilling, it is highly recommended to consider
pre-existing fractures on wellbore wall with which the lost circulation fractures may
not propagate perfectly along the direction of maximum horizontal stress as
assumed in this dissertation.
Thermal effect is an important factor to consider in the modeling studies of lost
circulation and wellbore strengthening.
For the study of preventive wellbore strengthening based on strengthening wellbore
with mudcake, dynamic filtration tests are recommended to investigate the time-
dependent developments of both external and internal mudcake, and then
incorporate the test results with advanced numerical models to simulate time-
dependent stress and failure around wellbore.
For the study of remedial wellbore strengthening, advanced numerical models that
have the capabilities of simulating transportation and deposition of LCMs in the
269
lost circulation fractures will be very useful for modeling the dynamic fracture
bridging/plugging process in wellbore strengthening.
For better application of wellbore strengthening techniques, it is important to
accurately and quickly estimate/measure the geometry of lost circulation fractures
during drilling for selecting/adjusting the size distribution of LCMs in a real-time
manner. Improved or new logging while drilling techniques are needed for
acquiring better knowledge of drilling-induced or pre-exiting natural fractures on
the wellbore wall.
The current wellbore strengthening studies mainly focus on addressing fluid loss
through hydraulically induced fractures in sand/shale formations. Severe losses are
also commonly encountered in carbonate formations with vugs, cavities, and large
fractures. More efforts on addressing lost circulation in carbonates are
recommended for further study.
270
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